QB 
8 1 

KG 

UC-NRLF 


THE  FIGURE 


EARTH. 

SECTION  I— HISTORICAL 

SECTION  II,— THE  OBLATE  SPHEROIDAL 
HYPOTHESIS. 


FRANK   C.  ROBERTS,  C.E. 


REPRINTED,  WITH  ADDITIONS, 
FROM  TAN  NOSTRAND'S  ENGINEERING  MAGAZINE. 


NEW  YOEK: 

;i).  VAN  NOSTRAND,   PUBLISHER, 
23  MURRAY  AND  27  WARREN  STREETS. 

1885. 


COIPTEIGHT     D.  VAN  NOSTRAND,    1886. 


PREFACE. 

IN  presenting  the  following  pages  to 
the  public  our  object  has  been  twofold. 

In  the  first  place,  we  have  endeavored 
to  place  before  our  readers  such  historical 
data,  in  connection  with  the  Figure  of 
the  Earth,  as  may  prove  of  an  interesting 
and  instructive  character. 

Secondly,  we  have  sought  to  arrange, 
in  a  compact  form,  the  important  mathe- 
matical principles  for  the  deduction  of  the 
Figure  of  the  Earth  upon  the  spheroidal 
hypothesis.  Hitherto  these  principles 
were  to  be  found  only  among  a  mass  of 
purely  mathematical  discussions  of  the 
Earth  in  the  fluid  state,  the  theory  of 
attractions,  etc. ;  and  in  omitting  these 
latter  in  the  present  treatise  we  hope 
to  have  simplified  the  study  of  the  sub- 
ject, and  thereby  to  have  supplied  a  want 
long  felt  by  students  of  Geodesy. 

770^ 


IV 

In  the  preparation  of  this  essay,  we 
have  had  access  to  works  of  standard  au- 
thors, and  we  cordially  acknowledge  our 
indebtedness  to  many,  but  more  espe- 
cially to  the  essays  of  Airy,  Pratt,  and 
Clarke. 

PRINCETON,  N.  J., 

March  12,  1885. 


SECTION   I. 

THE  FIGURE  OF  THE  EARTH, 

HISTORICAL. 


THE  FIGURE  OFTHE  EARTH 


SECTION  I.-HISTORICAL. 

CHAPTEK  I. 

EARLY  SUPPOSITIONS. 

THE  progress  of  the  science  of  astron- 
omy is  the  continued  triumph  of  the 
powers  of  the  intellect  over  the  first  er- 
roneous conceptions  of  the  senses ;  and 
its  history  is  so  allied  to  that  of  the  hu- 
man mind  that  we  cannot  help  feeling  a 
strong  inclination  to  know  at  what  time, 
and  by  what  people,  the  hypotheses  upon 
which  the  science  is  based  were  first  ad- 
vanced. 

The  value  of  an  hypothesis  is  estimated 
by  the  number  of  difficult  phenomena  it 
explains.  From  this  standpoint  few  sup- 
positions can  be  found  to  have  been  more 
important  than  that  which  assigned  to 


the  earth  its  approximate  figure  and  mag- 
nitude. 

Little  is  known  of  the  early  history  of 
this  hypothesis,  for  it  is  enveloped  in 
those  dark  ages  of  antiquity  when  the 
revolutions  of  empires  were  imperfectly 
recorded,  not  to  speak  of  the  calm  specu- 
lations of  quiet  and  thoughtful  men. 

To  its  first  inhabitants  the  earth  must 
have  appeared  as  an  extended  fixed  plane, 
the  extremities  of  which  apparently  sup- 
ported the  vast  dome  of  the  heavens. 
Among  the  ancients  the  prevailing  opin- 
ion was  that  the  surface  of  the  earth  was 
flat,  that  the  visible  horizon  was  the 
boundary  of  the  earth,  and  the  ocean  the 
boundary  of  the  horizon  ;  that  the  earth 
and  heavens  were  the  whole  visible  uni- 
verse, and  that  all  beneath  the  earth  was 
Hades. 

The  Chaldees  had  quite  an  opinion  of 
their  own  regarding  the  shape  of  the 
earth.  They  believed  it  to  have  the 
form  of  a  boat  turned  upside  down. 
This  theory  was  thoroughly  believed  in 
by  the  Chaldean  astronomers,  and  they 


9 


endeavored  to  support  it  by  scientific 
argument.  We  should  express  the  same 
idea,  at  present,  by  comparing  the  earth 
to  an  orange,  the  top  of  which  had  been 
cut  off,  leaving  the  orange  upright  upon 
the  flat  surface  thus  produced.* 

Evidently  ignorant  of  many  important 
physical  laws,  the  early  suppositions  of  the 
ancients  regarding  the  Figure  of  the  Earth 
are  in  a  great  measure  ludicrous.  Being 
but  imperfectly  acquainted  with  the  sci- 
ence of  astronomy,  and  their  observa- 
tions being  rude  and  inaccurate,  they 
were  led  to  base  their  theories  upon  false 
assumptions.  It  might  be  well  to  except 
from  this  sweeping  statement  the  theories 
of  the  early  Eastern  astronomers.  The 
earliest  astronomical  records  that  can.  be 
conceived  of  as  authentic  are  found  in 
China,  and  go  back  as  far  as  800  B.  C., 
when  we  find  eclipses  observed  and  reg- 
istered. This  would  naturally  lead  us  to 
conclude  that  the  observers  were  ac- 
quainted with  the  fact  that  the  sun  and 
heavenly  bodies  were  visible  to  people 
*  Lenormant's  Chaldean  Magic.  Page  150. 


10 


towards  the  east  sooner  than  people  to- 
wards the  west.  As  this  could  not  occur 
unless  the  earth  were  curved,  we  may  as- 
sume with  almost  a  certainty  that  the 
globular  figure  of  the  earth  was  known 
in  China  at  least  800  years  B.  C. 

Again,  astronomy  was  evidently  culti- 
vated as  a  science,  and  whatever  may  have 
been  their  suppositions  as  to  the  Figure 
of   the   Earth,  we   may  suppose  that  at 
the  first  they  were  led  to  conclude  that 
the  heavens  were  spherical  by  observing 
that  those  stars  sufficiently  elevated  to- 
wards the  north  pole,  performed  their  en- 
tire revolution  around  the  pole  without 
interruption  ;  from  which  it  might  by  an 
easy   inference   be    concluded   that    the 
other  stars,  though  concealed  from  view, 
pursued  their  course  in  the  same  man- 
ner.    When  once  the  theory  of  a  revolv- 
ing heaven  was  accepted,  it    would    be 
comparatively  an  easy  matter  to  conclude 
that  the  earth  was  globular. 

The  most  authentic  records  of  early  re- 
searches in  connection  with  the  Figure  of 
the  Earth  come  to  us  from  Greece.  The 


11 


discovery  that  the  earth  is  not  a  plane  is 
ascribed  to  Thales,  of  Miletus>  B.  C.,  640. 
Anaximander,  B.C.  570,  Anaxagoras,  B.C. 
460,  claimed  a  cylindrical  shape  for  the 
earth,  estimating  the  height  as  three  times 
the  diameter,  the  land  and  water  being  on 
the  upper  base.  Plato,  B.  C.  400,  called 
the  earth  a  cube.  Aristotle  is  commonly 
supposed  to  have  first  advanced  the 
theory  that  the  earth  was  spherical.  This 
supposition  we  consider  unwarranted. 

Egypt  in  the  days  of  early  Greece, 
is  known  to  have  taken  the  lead  in  all 
philosophical  pursuits.  Thales,  the 
first  Grecian  mathematician,  studied 
in  Egypt,  but  long  before  his  time, 
geometry,  astronomy,  and  other  sciences, 
had  been  held  in  high  repute.  Plato 
ascribed  the  invention  of  geometry  to 
Thoth ;  lamblicus  says  it  was  known  in 
Egypt  during  the  reign  of  the  gods,  and 
Manetho  attributes  a  knowledge  of 
science  and  literature  to  the  earliest 
kings.  These  reputed  scientific  atttain- 
ments  are  fully  verified  by  the  testimony 
of  the  ancient  monuments  of  Egypt.  If 


12 


we  are  to  credit  Eustathius,  the  Egypti- 
ans marked  their  lines  of  march  and  con- 
quest on  maps.  This  would,  of  course, 
necessitate  a  knowledge  of  mensuration 
and  surveying,  and  we  can  now  explain 
how  it'  was  that  the  Isrealites  were 
enabled  to  divide  the  land  of  Canaan 
when  commanded  to  do  so  by  Joshua.* 
It  was  this  known  progress  made  in  the 
branches  of  scientific  research,  that  in- 
duced the  Greeks  to  study  in  Egypt,  and 
later,  actuated  Pythagoras,  Eudoxus, 
Plato  and  Aristotle,  to  pay  extended 
visits  to  this  fountain  head  of  knowl- 
edge.f 

It  can  hardly  be  supposed  that  Pythag- 
oras suggested  without  previous  experi- 
ence the  Copernican  theory,  the  sun 
being  the  center  of  our  system  ;  or  the 
obliquity  of  the  ecliptic,  or  the  moon's 
"  borrowed  light ;  "  or  the  explanation 
of  the  milky  way  being  a  collection  of 
stars.  These  theories,  according  to 


*  Joshua  XVIII.    Kitto's  Daily  Bible  Illustrations. 
Page  286. 
t  Kawlinson's  Herodotus. 


13 


Aristotle,*  were  taught  by  Democritus, 
and  by  Anaxagoras,  the  former  of  whom 
studied  astronomy  for  five  years  in 
Egypt. 

The  same  may  be  said  of  the  principles 
by  which  the  heavenly  bodies  were  at- 
tracted to  a  center  and  impelled  in  their 
order,  the  theory  of  the  eclipses,  and  the 
proof  of  the  Earth  being  spherical.^ 
These  and  many  other  ideas  were  doubt- 
less borrowed  from  the  Egyptians  and 
Babylonians,  from  whose  early  discover- 
ies so  much  has  been  derived  concerning 
the  heavenly  bodies. $  From  these  state- 
ments it  seems  highly  probable,  that  the 
spherical  hypothesis  was  held  before  the 
time  of  Aristotle,  and  that  the  basis  for 
his  conclusions  in  regard  to  the  Figure  of 
the  Earth,  was  derived  from  his  knowl- 
edge of  the  science  of  astronomy  as 
taught  by  the  Egyptians.  Moreover, 
Cicero,  on  the  authority  of  Theophrastus, 
mentions  Hycetas,  of  Syracuse,  a  Pythag- 


*  Arist.  Met.  1.8. 

t  Aristotle  de  Coel,  11, 14. 

$  Aristotle  de  Coel,  11,  12. 


14 


orean,  who  claimed  that  the  earth  re- 
volved in  a  circle  around  its  own  axis. 
Aristotle  himself  also  observes,  that 
though  most  philosophers  say  the  earth 
is  the  center  of  the  system,*  "the 
Pythagoreans,  who  live  in  Italy,  claim 
that  fire  is  the  center,  and  the  earth 
being  one  of  the  planets  rotates  about 
the  center  and  makes  night  and  day.'' 

It  appears  also,  that  the  celebrated 
ancient  Hindo  astronomer,  Aryabhatta, 
maintained  the  doctrine  of  the  earth's 
diurnal  revolution  around  its  axis.f 
The  "  sphere  of  the  stars,"  he  affirms, 
"  is  stationary,"  and  the  "  earth  making 
a  revolution  produces  the  daily  rising 
and  setting  of  the  stars  and  planets." 
Now  it  is  evident  that  these  theories 
could  not  be  held  without  the  assump- 
tion of  a  spherical  earth,  and  we  there- 
fore conclude  that  the  spherical  hypothe- 
sis was  advanced  long  before  the  time  of 
Aristotle. 

The  progress  of  knowledge   concern- 

*  Aristotle  de  Coel,  11,  13. 

t  Researches  Bengal  Asiatic  Society.    Vol.  12. 


15 


ing  the  Figure  of  the  Earth  has  from  the 
earliest  times  been  closely  connected  with 
the  study  of  astronomy.  As  in  this  latter 
science,  first  impressions  are  abandoned, 
and  all  conclusions  are  in  striking  contra- 
diction to  those  of  superficial  observa- 
tion ;  so,  as  man  progressed,  the  earth 
became  divested  of  its  flattened  shape 
and  character  of  fixidity,  and  was  shown 
to  be  a  globular  body  turning  swiftly 
upon  its  own  axis,  and  moving  through 
space  with  great  rapidity. 

The  idea  of  the  earth  being  a  globe  is- 
now  so  familiar  to  us  that  arguments  in 
proof  of  it  are  almost  unnecessary.  Yet 
familiar  as  this  fact  now  is,  many  ages 
must  have  elapsed  before  it  was  univers- 
ally received.  So  difficult  was  it  to  con- 
ceive how  the  inhabitants  of  the  opposite 
hemisphere  could  exist  with  their  heads- 
down  wards,  that  we  find  St.  Augustine 
in  the  5th  century  vehemently  contend- 
ing against  the  possibility  of  the  exist- 
ence of  an  antipodes. 


16 

CHAPTER  II. 

THE  SPHERICAL  HYPOTHESIS. 

It  may  be  well  to  understand  at  this 
point  that  when  we  speak  of  the  Figure 
of  the  Earth,  we  mean  the  figure  which 
would  be  assumed  by  the  earth  were  it 
covered  entirely  with  water,  and  more 
specifically  water  at  mean  sea  level.*  As 
for  the  inequalities  on  the  surface  of  the 
«arth  owing  to  mountains  and  valleys, 
they  are  of  no  moment  in  the  estimation 
of  the  general  figure,  being  of  much  less 
account  with  respect  to  relative  propor- 
tion than  the  asperities  on  the  surface  of 
an  orange  with  regard  to  the  orange  it- 
self. Upon  an  artificial  globe  of  6f  feet 
in  diameter,  Mount  Chimborazof  would 
be  represented  by  a  grain  of  sand  less 
than  1-20  of  an  inch  in  thickness. 

The  active  curiosity  of  man  did   not 
rest  contented  with  having  assumed  that 

*  Scientifically  speaking  the  figure  of  the  earth  is 
interpreted  as  meaning  the  mean  surface  of  the  sea 
imagined  to  percolate  the  continents  by  canals. 

t|21,424  feet  above  mean  tide. 


17 


the  earth  was  a  sphere,  but  proceeded 
to  ascertain  the  exact  dimensions  of  the 
planet.  In  the  course  of  his  discussion 
Aristotle  states,  as  does  Archimedes  (B. 
C.  250),  that  mathematicians  estimated 
the  circumference  of  the  earth  at  300,000 
stadia. 

The  first  approximation  to  the  magni- 
tude of  the  earth,  however  inaccurate, 
must  have  been  at  that  time  a  most  im- 
portant addition  to  the  stock  of  natu- 
ral knowledge.  Indeed,  except  with  a 
view  to  some  very  refined  scientific  in- 
vestigation, the  general  idea  which  the 
ancients  had  of  the  magnitude  of  the 
earth  differs  but  little  from  that  of  the 
moderns  ;  for  we  are  so  incapable  of  the 
appreciation  of  number  or  magnitude 
when  either  exceeds  a  certain  limit,  that 
the  difference  between  their  results  and 
ours,  makes  little  or  no  difference  in  the 
general  idea  which  we  hold  as  to  the  size 
of  the  earth. 

Eratosthenes,  B.  C.  230,  was  apparent- 
ly the  first  to  conceive  a  method  for  the 
deduction  of  the  length  of  the  circumfer- 


18 


ence  of  the  earth.  Although  his  results 
are  probably  sadly  inaccurate,  the  method 
which  he  adopted  is  in  essence  identical 
with  that  followed  at  the  present  time. 

As  it  is  impossible  for  us  to  occupy  *a 
position  from  which  the  earth  may  be 
viewed  as  a  whole  and  compared  with 
some  standard  of  measure,  we  are  com- 
pelled to  resort  to  geometrical  principles 
in  the  determination  of  its  figure  and 
magnitude.  The  problem  is  rendered 
more  difficult  by  the  fact  of  there  being 
no  fixed  landmarks  or  standard  lines 
upon  the  surface  of  the  earth,  indicating 
aliquot  parts  of  the  earth's  circumfer- 
ence. It  therefore  becomes  necessary 
to  refer  our  situation  on  the  earth  to 
objects  external  to  our  own  planet. 
Such  marks  are  afforded  by  the  heavenly 
bodies. 

By  observations  of  the  meridian  alti- 
tudes of  stars,  and  from  their  known 
polar  distances,  we  determine  the  altitude 
of  the  pole-star,  which  is  also  the  lati- 
tude of  the  place  at  which  the  observa- 
tions are  made. 


19 

Let  us  suppose  then,  that  we  wish  to 
determine  the  length,  on  the  surface,  of 
one  degree  of  the  earth's  circumference. 
Let  us  suppose  also  that  we  know  the 
distance  between  two  places  on  the  same 
meridian.  Then  having  determined  the 
latitudes  of  the  two  places,  their  differ- 
ence may  be  taken  as  representing  the 
angle  at  the  center  of  the  earth  corre- 
sponding to  the  measured  distance  on 
the  surface.  Dividing  the  distance  by 
the  angle,  we  find  the  length  of  a  merid- 
ian arc  equivalent  to  one  degree  of  lati- 
tude, and  this  multiplied  by  360  gives 
us  the  length  of  the  earth's  circumfer- 
ence. 

Where  local  difficulties  compel  the  ob- 
servers to  deviate,  in  the  measurement  of 
the  distance,  from  the  line  of  the  merid- 
ian, the  amount  of  the  deviation  must  be 
noted.  A  very  simple  calculation  will 
enable  us  to  reduce  the  measured  dis- 
tance to  the  corresponding  length  on  the 
meridian.  It  seems  hardly  necessary  to 
add  that  this  measurement  must  be 
made  with  the  greatest  care  and  accuracy,. 


20 


for  an  error  in  the  measured  length  of 
one  degree  is  multiplied  360  times  in  the 
circumference,  and  nearly  58  times  in  the 
deduced  radius  of  the  earth. 

Such  in  its  simplest  form  is  the  geodetic 
operation  called  the  measurement  of  an  arc 
of  a  meridian,  and  is  in  essence  the  method 
employee!  by  Eratosthenes.  He  knew 
that  at  Syene,  in  S.  Egypt,  on  the  day  of 
the  summer  solstice  at  mid-day,  objects 
cast  no  shadows,  whence  he  concluded 
that  the  sun  was  in  the  zenith.  In  Alex- 
andria, at  the  same  periocf,  he  observed 
that  the  sun  made  an  angle  with  the  ver- 
tical of  7°  12',  or  -g*¥th  of  a  circumference. 
Assuming  Alexandria  to  be  directly  north 
of  Syene,*  he  concluded  the  length  of 
the  circumference  to  be  50  times  the  dis- 
tance between  these  two  places,  or  250,000 
stadia.  Of  course  this  determination  was 
very  imperfect,  for  with  the  instruments 
of  his  time  he  was  compelled  to  neglect 
the  diameter  of  the  sun  in  the  determin- 
ation of  declination.  This  occasioned 

*  The  error  of  this  assumption  was  about  three  de- 
grees. 


21 


an  error  of  J°  in  the  length  of  the  celes- 
tial arc  at  Syene.  The  measurement  of 
the  distance  between  Syene  and  Alex- 
andria was  probably  also  very  ina,ccu- 
rate. 

The  next  attempt  to  solve  the  problem 
was  made  by  Posidonius,  B.  C.  90.  In- 
stead of  using  the  sun  for  the  determin- 
ation of  the  difference  of  latitude,  he 
found  the  celestial  arc  by  means  of  the 
star  Ganopus.  At  Rhodes  this  star  when 
on  the  meridian  is  just  visible  above  the 
horizon,  while  at  Alexandria  its  meridian 
altitude  is  7°  30'.  The  distance  between 
the  two  places  being  known,  he  deduced 
240,000  stadia  as  the  circumference  of 
the  earth. 

Ptolemy,  an  astronomer,  A.  D.  160,  in 
his  treatise  on  Geography,  gives  500 
stadia  as  the  length  of  a  degree.  This 
value  would  give  for  total  length  of  the 
circumference  180,000  stadia — a  result 
widely  different  from  any  previously  de- 
duced. 

Unfortunately  the  degree  of  approxi- 
mation attained  in  these  results  cannot 


be  known,  as  we  have  no  value  for  the 
length  of  the  stadium.  It  probably  had 
different  values  dependent  upon  time  and 
place. 

In  the  year  819  the  Caliph  Almamoun 
caused  the  astronomers  of  Bagdad  to 
measure  an  arc  of  the  meridian  on  the 
plains  of  Mesopotamia  by  means  of 
wooden  rods.  Authorities  differ  as  to 
the  resulting  deduction  of  the  length  of 
a  degree ;  some  claiming  that,  failing  in 
their  own  observations  owing  to  insur- 
mountable obstacles,  the  Arabians  adopt- 
ed the  result  of  the  Grecian  astronomer 
Ptolemy.  Others  hold  that  they  found 
for  the  length  of  a  degree  56f  Arabian 
miles,  or  approximately  71  English  miles. 

From  this  time  until  the  revival  of  let- 
ters, interest  in  this  subject  seems  to  have 
disappeared.  Speculation  was  at  a  stand- 
still for  700  years.  Former  theories  and 
suppositions  were  forgotten  and  lost  amid 
the  social  storms  of  the  middle  ages.  Man 
was  again  ignorant  of  the  form  and  dimen- 
sions of  the  planet  which  had  been  as- 


23 

signed    to    him    in    the    immensity    of 
space. 

Early  in  the  15th  century  the  question 
of  the  form  of  the  earth  began  a  second 
time  to  attract  the  attention  of  thought- 
ful men.  The  prevalent  idea  was  that 
the  earth  was  a  plane.  This  time  it  was 
not  the  philosophers  but  the  navigators 
who  looked  with  doubt  upon  this  sup- 
position. Columbus  fearlessly  asserted 
the  earth  to  be  globular,  and  after  the 
voyage  of  Magellan  around  the  earth,  the 
globular  hypothesis  was  once  more  ac- 
cepfced.  Immediately  endeavors  were 
made  to  determine  the  size  of  the  earth. 
In  1525  Fernel  made  a  determination  of 
the  length  of  a  degree  by  deducing  the 
difference  of  latitude  between  Paris  and 
Amiens,  and  measuring  the  distance  by 
observing  the  number  of  revolutions 
made  by  his  coach  wheel  in  traveling 
from  one  place  to  the  other.  From  these 
observations  he  deduced  the  length  of 
one  degree  to  be  57,050  toises  or  364,960 
English  feet.  (The  toise — an  old  French 
measure — is  practically  equal  to  1.949 
meters,  or  6.3946  English  feet.) 


24 


In  1617  Willebrord  Snell  conceived 
the  idea  of  the  deduction  of  the  length 
of  distances  by  means  of  a  series  of  tri- 
angles measured  from  a  known  base.  This 
was  the  first  instance  of  the  application 
of  the  invaluable  principle  of  trigono- 
metrical surveying  which,  since  that  time, 
has  become  general  in  all  extensive  sur- 
veys. 

Snell  measured  his  base  line  upon  the 
frozen  surface  of  the  meadows  between 
Leyden  and  Soeterwood.  The  angles 
he  measured  by  means  of  a  quadrant  of 
5J  feet  radius .  His  result  for  the  length 
of  a  degree  was  55,020  toises,  or  approxi- 
mately 66.63  miles. 

In  1633  Norwood,  in  England,  adopt- 
ing a  method  similar  to  that  of  Fernelrs 
deduced  57,424  toises  for  the  length  of  a 
degree. 

We  are  now  brought  down  to  the  time 
of  Picard,  whose  invaluable  adaptation 
of  the  telescope  to  circular  instruments 
for  measuring  angles,  marks  an  era  in  the 
progress  of  geodetic  science.  Hitherto 
the  measurement  of  angles  was  roughly 


25 


made  by  the  use  of  sights  similar,  only 
much  more  unreliable,  to  those  used  on 
rifles.  Picard  first  introduced  spider  lines 
in  the  focus  of  the  telescope,  whereby 
a  far  higher  degree  of  precision  in  de- 
termination of  the  position  of  a  distant 
point  may  be  obtained,  by  covering  the 
point  in  question  with  the  intersection  of 
spider  lines,  which  is  so  placed  as  to  be 
exactly  in  the  center  line  of  the  telescope. 
In  his  determination  of  the  length  of  a  de- 
gree he  used  the  trigonometrical  method, 
measuring  twice  a  base  line  of  nearly 
seven  miles  in  length.  His  measurement 
is  the  first  executed  with  anything  like 
scientific  precision.  He  even  calculated 
the  error  produced  by  his  instrument  be- 
ing out  of  the  center  of  his  station,  and 
determined  his  difference  of  latitude  by 
means  of  the  zenith  distance  of  a  star  in 
Cassiopeia  measured  with  a  sector.  At 
this  time  the  effect  of  aberration,  refrac- 
tion and  nutation  were  unknown,  never- 
theless his  result  of  57,060  toises,  or 
nearly  69.76  miles  is  marvelously  near 
that  of  later  determinations. 


26 


The  measurement  of  an  arc  of  the 
meridian,  although  the  most  reliable,  is 
not  the  only  method  by  which  the  Figure 
of  the  Earth  upon  the  spherical  hypoth- 
esis may  be  deduced.  One  simple  ex- 
pedient consists  in  determining  the  dip 
or  angle  of  depression  of  the  horizon. 
Take,  for  instance,  the  case  of  a  mountain 
near  the  sea  coast..  Knowing  the  height 
of  the  mountain  above  the  sea,  and  the 
angle  of  depression  from  its  top  to  the 
horizon,  we  can,  by  an  easy  mathematical 
formula,  deduce  the  following  equation 
in  which  r  is  the  radius  of  the  earth,  h 
the  height  of  the  mountain,  and  d  the 
distance  from  the  mountain  to  the  hori- 
izon: 


This  principle  was  applied  more  than 
200  years  ago  at  Mount  Edgecombe,  and 
since  that  time  at  Ben  Nevis. 

Or,  again,  we  may  by  the  application 
of  the  following  proposition  form  a  pro- 
portion from  which  the  diameter  of  the 
earth  may  be  found  —  the  earth's  diameter 
bears  the  same  proportion  to  the  dis- 


27 


tance  of  the  visible  horizon  from  the  eye 
as  that  distance  does  to  the  height  of 
the  eye  above  the  sea  level. 

Both  these  methods,  of  course,  only 
furnish  means  of  determining  the  size  of 
the  earth  with  a  rough  approximation. 
Refraction  bends  the  visual  lines  out  of 
the  truly  rectilinear  direction,  and,  there- 
fore, introduces  a  serious  error  in  the 
result. 


28 


CHAPTER  HI. 

SPHEROIDAL  AND  ELLIPSOIDAL  HYPOTHESES. 

Up  to  1690  astronomers  supposed  the 
form  of  the  earth  to  be  nearly  that  of  a 
perfect  sphere,  and  consequently  the 
length  of  degrees  in  all  latitudes  pre- 
cisely equal. 

In  1690-1718,  J.  and  D.  Cassini  pub- 
lished results  showing,  that  although  the 
measures  of  meridional  arcs  made  in  va- 
rious parts  of  the  globe  agreed  suffi- 
ciently to  prove  that  the  supposition  of  a 
spherical  figure  is  not  very  remote  from 
the  truth,  yet  exhibited  discordances  far 
greater  than  could  be  attributable  to  er- 
rors of  observation,  and  which  rendered 
it  evident  that  the  spherical  hypothesis 
was  untenable.  Immediately  upon  this 
discovery,  new  interest  was  awakened  in 
the  subject.  The  works  of  previous  sci- 
entists upon  this  subject  were  carefully 
examined,  and,  as  a  result,  it  became 
known  that  Picard,  as  early  as  1671,  in 
his  work  on  the  Figure  of  the  Earth, 


29 


mentions  a  conjecture  proposed  to  the 
French  Academy  that,  supposing  the  di- 
urnal motion  of  the  earth,  heavy  bodies 
should  descend  with  less  force  at  the 
equator  than  at  the  poles  ;  and  that,  for 
the  same  reason,  there  should  be  a  varia- 
tion in  the  length  of  the  pendulum  vi- 
brating seconds  in  different  latitudes,  for 
the  time  of  oscillation  of  a  pendulum  of 
constant  length  depends  upon  the  intens- 
ity of  the  force  of  gravity. 

In  the  same  year  Richer  was  sent  to 
Cayenne,  in  equatorial  S.  A.,  and  was 
especially  charged  by  the  Academy  to  ob- 
serve the  length  of  the  pendulum  vi- 
brating seconds.  On  his  return  he  stated 
that  the  difference  between  the  seconds 
pendulum  at  Paris  and  Cayenne  was  one 
line  and  a  quarter,  that  at  Cayenne  be- 
ing the  shorter.  Moreover,  the  clock 
which  Richer  took  to  Cayenne,  having 
been  adjusted  to  beat  seconds  at  Paris, 
retarded  two  minutes  a  day  at  Cay- 
enne, so  that  no  doubt  remained  of  the 
diminution  of  the  force  of  gravity  at  the 


30 


equator.*  This,  as  it  was  the  first  direct 
proof  of  the  diurnal  motion  of  the  earth, 
was  also  what  led  Huygens  to  suspect 
that  there  was  a  protuberance  of 
the  equatorial  parts  of  the  earth,  and 
a  corresponding  depression  of  the  poles. 
Cassini  had  already  observed  this  phe- 
nomenon in  the  figure  of  Jupiter,  which 
analogy  strongly  favored  the  supposition 
of  a  similar  peculiarity  in  the  shape  of 
the  earth,  f  Since,  then,  it  was  evident 
that  the  meridian  section  of  the  earth  was 
not  a  circle,  what  was  the  next  simplest 
supposition  that  could  be  made  respect- 
ing the  nature  of  the  meridian.  In  the 
flattening  of  a  round  figure  at  two  oppo- 
site points,  and  its  protuberance  at  points 
rectangularly  situated  to  the  former,  we 
recognize  the  distinguishing  feature  of 

*  See  Newton's  Principia,  Book  III. 

t  The  difference  of  the  diameters  of  Jupiter  amounts 
almost  to  l-10th,  and  when  we  compare  the  exact 
measure  of  this  depression,  the  dimensions  of  Jupiter 
and  the  time  of  his  rotation,  with  like  phenomena  con- 
nected with  the  earth,  we  find  for  this  latter  planet  a 
proportional  depression  of  l-388th,  which  is  very  near- 
ly identical  with  the  value  deduced  from  the  great 
French  measurement. 


31 


the  elliptic  form.  Thus  mathematicians, 
after  discarding  the  spherical  hypothesis 
assumed  the  meridian  to  be  an  ellipse. 
The  geometrical  properties  of  that  curve 
enabled  them  to  assign  the  proportion  be- 
tween the  lengths  of  the  axes  which 
would  correspond  to  any  proposed  rate 
of  variation  in  its  curvature,  as  well  as 
to  fix  upon  the  absolute  lengths  corre- 
sponding to  any  assigned  length  of  a  de- 
gree in  a  given  latitude. 

Spheroids  are  generated  by  the  revolu- 
tion of  ellipses  about  one  or  another  of 
their  axes.  Every  ellipse  has  two  axes, 
one  passing  through  the  foci  is  called  the 
major  axis,  while  the  other — perpendicu- 
lar to  the  major  axis  at  its  middle  point 
— is  called  the  minor  axis.  When  the  el- 
lipse revolves  about  its  minor  axis  it  gen- 
erates what  is  called  an  oblate  spheroid, 
and  when  it  revolves  about  its  major 
axis  the  figure  generated  is  named  a  pro- 
late spheroid.  The  ellipticity  is  the 
amount  of  variation  of  the  form  of  the 
spheroid  from  a  sphere  of  like  content,  or 
the  amount  of  flattening  at  the  poles* 


32 


This  is  expressed  by  dividing  the  differ- 
ence of  the  semi-major  and  minor  axes 
by  semi-major  axis.  The  eccentricity  of 
an  ellipse  is  equal  to  the  distance  from 
the  center  to  one  of  its  foci  divided  by 
the  semi-major  axis. 

Huygens  was  the  first  person  who  at- 
tempted to  determine  the  Figure  of  the 
Earth  by  direct  calculation,  but  in  his  in- 
vestigation he  assumes  that  the  whole  of 
the  attractive  force  resides  in  the  center 
of  the  earth,  and  that  its  power  varies  as 
the  square  of  the  distance.  This  hypoth- 
esis, since  the  discovery  of  the  law  of 
universal  gravitation,  has  been  found  in- 
admissible, and  therefore  his  results  were 
largely  in  error. 

In  the  course  of  the  discussion  of  Cas- 
sini's  observation  of  the  variation  in 
length  of  the  seconds'  pendulum  in  differ- 
ent latitudes,  and  consequent  diminution 
of  the  force  of  gravity  at  the  equator,  it 
was  claimed  that  this  diminution  might 
be  due  to  the  counteracting  effect  of  the 
centrifugal  force  occasioned  by  the  rota- 


33 


tion  of  the  earth.  Newton*  showed  that 
even  after  making  allowances  for  this  ef- 
fect, the  difference  between  the  force  of 
gravity  at  Paris  and  Cayenne  was  too 
great  for  the  spherical  hypothesis,  and 
further,  upon  the  assumption  that  the 
earth  is  a  homogeneous  fluid,  and  sup- 
posing its  density  to  be  the  same  through- 
out the  whole  mass,  and  assuming  that 
the  constituent  molecules  attract  one 
another  in  proportion  to  the  inverse 
square  of  the  distance,  he  demonstrated 
that,  in  consequence  of  rotation,  the 
earth  would  assume  the  form  of  an  oblate 
spheroid,  whose  ellipticity  would  amount 
to  ^th. 

Clairaut  was  the  first  to  advance  a  gen- 
eral solution  of  this  problem  adapted  to, 
the  hypothesis  of  a  variable  density.  He 
proved  that,  if  the  density  of  the  strata 
of  which  the  earth  is  composed  increases 
towards  the  center,  the  ellipticity  will  be 
less  than  in  the  hypothesis  of  Newton, 
and  greater  than  in  that  of  Huygens ; 
and,  again,  that  the  sum  of  the  fraction 

*  Pi-incipia,  Book  III. 


34 


representing  the  ellipticity  and  the  frac- 
tion expressing  the  augmentation  of  grav- 
ity at  the  poles  will  always  make  \  r  con- 
stant quantity,  which  is  equal  to  |  of  the 
fraction  which  expresses  the  proportion 
which  exists  between  the  centrifugal 
force  and  gravity  at  the  equator.  It  is 
by  means  of  this  theorem  we  are  enabled 
to  ascertain  the  Figure  of  the  Earth  by 
pendulum  experiments. 

These  theoretical  determinations  of 
Huygens,  Newton,  and  Clairaut  were,  up- 
on the  completion  of  surveys  made  by 
Cassini  in  France,  found  to  be  at  variance 
with  his  results.  He  found  the  length  of 
one  degree  of  a  meridian  south  of  Paris 
to  be  57.092  toises,  while  north  of  the 
city  it  was  only  56.960  toises.  This  led 
to  the  conclusion  that  the  earth  is  a  pro- 
late spheroid.  Here,  of  course  was  ma- 
terial for  a  controversy ;  in  view  of  this 
fact  the  French  Academy  sent  out  two  ex- 
peditions to  make  measurements  that 
would  definitely  settle  the  matter.  These 
expeditions  set  out  in  1735  ;  Bouguer, 
Godin  and  La  Condamine  proceeded  to 


35 


Peru,  and  after  ten  years'  work  they 
measured  an  arc  of  above  3°  between  the 
parcels  2'  31"  N.,  and  3°  4'  32"  S. 
latitude.  Maupertius,  Clairaut,  Camas 
and  Le  Monnier,  arriving  in  Lapland, 
measured  an  arc  of  57  minutes,  and  re- 
turned within  16  months.  The  results 
deduced  from  these  observations  con- 
curred in  proving  that  the  degrees  of  the 
meridian  increase  very  sensibly  in  length 
from  the  equator  to  the  high  latitudes, 
and  from  this  time  dates  the  undisputed 
conclusion  that  the  a  earth  is  an  oblate 
spheroid,  rather  than  a  sphere  or  prolate 
spheroid. 

The  deviation  from  the  spherical  form 
is  evidently  very  slight,  the  difference  be- 
tween the  equatorial  and  polar  diameter 
being  only  27  miles.  As  an  illustration, 
on  a  globe  24  inches  in  equatorial  diam_ 
eter,  and  on  which  the  thickness  of  a 
sheet  of  writing  paper  would  represent 
the  elevation  of  the  lands  above  the 
waters,  the  polar  axis  would  be  23.928 
inches,  or  in  other  words,  the  difference  be- 
tween the  polar  and  equatorial  axes  would 


36 


be  but  one-fourteenth  of  an  inch.  For 
this  reason  the  spherical  hypothesis  is 
sufficiently  accurate  for  many  purposes. 
When  this  hypothesis  is  used  in  geodeti- 
cal  operations,  the  radius  of  the  earth  as 
a  sphere  is  taken  as  the  average  of  all 
the  radii  of  the  spheroid.  This  radius  is 
equal  to  6.370  kilometers,  or  3.958  miles. 
In  the  ^determination  of  the  mean  length 
of  an  arc  of  1  °,  -g^-  of  the  length  of  an  el- 
liptical quadrant  of  the  spheroid  is  taken. 
Yarious  values  for  this  quadrant  have 
been  computed  by  different  mathemati- 
cians. The  one  deduced  by  Bessel  has 
been  in  long  use  for  geodetical  computa- 
tions, and  is  very  nearly  the  mean  of  the 
values  found  by  other  investigators. 
The  mean  length  of  one  degree  is,  ac- 
cording to  Bessel,  111.121  meters,  or 
69.043  miles. 

It  is  thus  seen  that  when  the  spherical 
hypothesis  is  applied  the  assumed  sphere 
is  one  having  an  equal  volume  with  that 
of  the  oblate  spheroid. 

There  is,  however,  in  these  values,  a 
serious  inconsistency,  for  the  quadrant  of 


37 


a  circle  corresponding  to  the  above  mean 
radius  is  nearly  6  kilometers  greater  than 
Bessel's  value  used  in  the  above  deter- 
mination of  the  length  of  1°.  For  this 
reason  the  value  sometimes  used  for  the 
radius  is  that  of  a  circle  whose  circum- 
ference is  equal  to  the  circumference  of  a 
meridian  ellipse,  or  3.956  miles— 6.367 
kilometers.  This  value  is  3  kilometers 
too  small,  but  the  error  is  unavoid- 
able. 

That  the  science  of  the  mathematicians 
had  described  in  a  general  way  the  fig- 
ure of  our  globe,  was  sufficient  to  satisfy 
the  curiosity  of  the  ordinary  individual, 
but  not  the  zeal  of  scientists  for  exact 
knowledge ;  they  further  endeavored  i^o 
obtain  the  precise  amount  of  the  depres- 
sion at  the  poles,  whose  existence  had 
been  proven  by  so  many  experiments. 
Material  was  accumulated,  new  arcs  were 
measured,  but  the  difficulty  of  an  exact 
determination  only  increased.  The  dif- 
ferent measures  of  degree  lengths  gave 
varying  values  for  this  depression  upon 
the  oblate- spheroidal  hypothesis.  An 


38 


Italian  mathematician,  named  Frisi, 
showed  the  variation  of  the  calculated  de- 
pression very  clearly  by  a  comparison  of 
the  measures  then  known.  The  follow- 
ing is  a  list  of  the  arcs  used  by  him  in  his 
computations,  and  also  of  the  astrono- 
mers to  whom  we  are  indebted  for  their 
determination  : 


39 


ill 


o>    . 

T3T3 

2  2 


ii 


i  i  i  i 


f 
I 

o 


. 
.0 


. 


40 


Frisi,  in  his  calculations,  sought  to  de- 
termine, according  to  Newton's  theory, 
the  data  for  a  regular  curve  from  which 
could  be  derived  the  above  values.  In 
this  he  was  unsuccessful.  The  curves 
were  either  too  large  or  too  small. 

The  values  for  the  ellipticity  of  the 
earth's  meridian  deduced  from  the  sur- 
veys instituted  by  the  French  Academy 
are  as  follows  : 


Lapland  and  French  arcs, 
Lapland  and  Peruvian  arcs,  -3^0  th  ; 
French  and  Peruvian  arcs, 


There  was  evidently  a  serious  discrep- 
ancy either  in  the  assumption  as  to  the 
form  of  the  earth,  or  in  the  accuracy  of 
the  determinations,  for  if  the  earth  were 
a  spheroid  of  revolution  these  results 
should  be  identical.  Following  this  dis- 
covery numerous  measurements  of  arcs 
of  meridian  were  made  in  different  parts 
of  the  world.  The  most  important  of 
these,  however,  were  executed  under  the 
direction  of  the  French  Government  in 
the  determination  of  the  length  of  the 


41 

meter — taken  as  one  ten-millionth  part  of 
the  quadrant  of  the  earth's  meridian. 
These  latter  observations  when  combined 
with  the  corresponding  values  in  the 
Peruvian  arc  gave  for  the  ellipticity, 
-g-g^th.  The  nearest  approximation  of 
the  calculated  curve  to  an  ellipse  whose 
minor  axis  would  be  to  its  major  in  the 
ratio  of  230  to  231  involved  an  error  of 
more  than  100  toises  to  the  degree. 
Frisi  then  determined  the  mean  value  for 
the  various  depressions  resulting  from 
the  above  data  and  found,  for  the  mean 
term,  a  depression  almost  identical  with 
that  furnished  by  the  observations  of  the 
pendulum  and  the  measurements  for  the 
determination  of  the  French  measures. 

The  evident  impossibility  of  finding  a 
regular  curve  to  correspond  to  the  dif- 
ferent degrees  measured,  gave  rise  to 
doubts  as  to  the  possibility  of  measuring 
a  degree  of  the  meridian  with  accuracy. 
The  instruments  then  employed  in  the 
determinations  were  liable  to  errors  of 
three  or  four  seconds  for  the  celestial 


42 


arc,    or   60    toises   for   a  terrestrial   de- 
gree.* 

The  attraction  of  mountains  upon  the 
plumb  line,  causing  a  deviation  of  the 
vertical,  was  another  source  of  er- 
ror. Thus,  if  the  direction  of  the  plumb 
line  at  the  extremities  of  the  arc  meas- 
ured deviated  from  the  normal  by  15  sec., 
it  would  cause  an  error  of  500  toises  or 
533  fathoms  in  the  final  result,  a  quantity 
greater  than  the  presumed  difference  of 
the  two  extreme  degrees  under  the  equa^ 
tor  and  the  pole.f  Towards  the  end 
of  the  last  century  various  attempts 
were  made  to  reconcile  the  accumu- 
lating data  with  the  spheroidal  hypoth- 
esis. Among  the  most  prominent  investi- 
gations are  those  of  Boscovich  in  1760r 
and  Laplace  in  1793  and  1799.  Laplace 
took,  as  the  basis  of  his  combinations, 
nine  of  the  measurements  used  by  FrisL 
The  curve  which  he  calculated  gave  for 
the  length  of  a  degree  a  value  too  small 
by  137.7  toises  (= nearly  268  meters),  or 
approximately  nine  seconds  of  latitude. 

*  Bouguer,  Fig.  de  la  Terre,  sect.  1,  §  4. 
t  Malt-Brun.  p.  25. 


43 

These  errors,  sajs  Laplace,  are  too  great 
to  be  admitted,  and  it  must  be  concluded 
that  the  earth  deviates  materially  from 
the  elliptical  figure.* 

When  compared  with  the  great  size  of 
the  earth,  this  deviation  of  the  figure  of 
the  earth  from  the  oblate  spheroidal 
form  is  very  slight.  As  previously  stated, 
for  many  practical  problems  it  is  suffi- 
ciently accurate  to  consider  the  earth  as 
a  sphere,  but  where,  for  the  purposes  of 
science,  it  is  necessary  to  apply  the  spher- 
oidal hypothesis,  mathematicians  have 
deemed  it  expedient  to  determine  the  ele- 
ments of  an  ellipse  agreeing  as  nearly  as 
possible  with  the  actual  meridian  section 
of  the  earth,  and  to  base  their  calcula- 
tions upon  the  resulting  spheroid.  . 

In  1805  Legendre  announced  the  meth- 
od of  least  squares  for  the  adjustment  of 
observations,  and  during  the  present  cen- 
tury numerous  applications  of  this  prin- 
ciple in  the  determination  of  the  mean  el- 
lipse of  the  earth's  meridian  have  been 
made,  the  principal  of  which  are  given  in 
the  table  on  page  44. f 

*  Hist.  Acad.  Paris,  1789.  t  Jordan. 


§^S?5?Qpioioo6ci' 


'GJQOlOiaoOCiOCo" 
1  V*  WJ  CS  O^  OO  OO  QO  Gti 
'G^C^CiCQOiOJC^XTi 


45 


Of  these,  the  values  of  Bessel  and 
Clarke  are  considered  the  most  reliable, 
and  the  spheroids  deduced  from  the  ele- 
ments calculated  by  these  investigators 
are  called  respectively  the  Bessel  and 
Clarke  spheroids.  The  dimensions  of 
the  terrestrial  spheroid  deduced  by  Bes- 
sel are  as  follows : 

Greater  or  equatorial  diameter,  7925.604 
miles. 

Lesser,  or  polar  diameter,  7899. 114  miles. 

Difference  of  diameters,  or  polar  com- 
pression, 26.491  miles. 

Proportion  of  diameters  as  299.15  to 
298.15. 

Probably  the  value  for  the  ellipticity 
deduced  from  pendulum  experiments*  is 
nearer  the  truth  than  any  deduced  from 
geodetic  data.  The  latter  values  have 
been  continually  approaching  those  of  the 
former,  and  we  have  every  reason  to  be- 
lieve that  when  perfection  of  geodetic  op- 
erations is  more  nearly  approached,  the 
results  will  be  practically  identical.  We 
give  below  the  elements  of  the  earth's 


46 

figure  deduced  from  pendulum  observa- 
tions : 

Ellipticity  =  -S55-T-.  Eccentricity ^^th. 

^OO.O  L& 

Quadrant  of  E's  Meridian  section =10001 
kilometers,  or  6214.62  statute  miles. 

In  1859  Gen.  de  Schubert,  in  attempt- 
ing to  find  a  continuous  curve  for  the 
meridian  which  would  satisfy  all  meas- 
ured geodetic  arcs,  suggested  the  hy- 
pothesis of  an  elliptic  equator  and  an  el- 
lipsoidal figure.  The  ellipsoid  is  not  a 
figure  of  revolution.  The  meridian  sec- 
tions, as  in  the  spheroid,  are  ellipses,  but 
the  equator,  instead  of  being  a  circle,  is 
an  ellipse.  The  curves  of  latitude,  how- 
ever, except  the  equator,  are  not  plane 
curves,  and  consequently  not  true  paral- 
lels. 

Thus,  we  see  that  the  ellipsoid  has 
three  unequal  axes  at  right  angles  to  each 
other. 

Gen.  de  Schubert  embodied  his  idea  in 
his  "Essai  d'une  Determination  de  la 
veritable  Figure  de  la  Terre,"  and  de- 


47 


duced  from  eight  meridian  arcs  an  ellips- 
oid of  the  following  elements  :* 

tf, =6,378,566  metres 

^2=6,377,837      " 

b  =6,356,719      " 

^1=2927l    f*=lm'       ^8881 
where  a}  «2  are  the  semi-equat.  axes.  b  = 
semi- polar  axis,  /,  /'.,=the  ellipticities  of 
the  greatest  and  least  meridian  ellipses, 
F  =  the  ellipticity  of  the  equator. 

In  1860  and  3,866  similar  calculations 
were  made  by  Capt.  A.  E.  Clarke.  Sub- 
sequent investigations  led  Clarke  in  1878 
to  publish  the  results  of  a  third  discus- 
sion, giving  as  the  elements  of  the  ellips- 
oid the  following : 

at  =  20,926,629  feet. 

a,  =  20,925,105     " 

£=20,854,477     " 

111 
^  =  oH7v  /'=296 73'  F  =  Wm 


The  present  opinion  in  regard  to  the 

*Mem.  d  TAcad.  Imp.  des  Sciences  de  St.  Peters 
bourg,  VII.  Serie,  Tome  1,  No.  6. 


48 


ellipsoidal  hypothesis  is  that  until  data  of 
a  more  general  and  accurate  kind  have 
been  accumulated,  the  elements  of  a  sat- 
isfactory ellipsoid  cannot  be  computed. 
Arcs  of  longitude  are  needed,  for  the  el- 
lipticities  of  the  meridians  differ  by  such 
small  quantities,  that  measurements  in 
their  directions  alone,  are  insufficient  to 
determine  with  much  precision  the  form 
of  the  equator  and  parallels. 

Aside  from  this,  the  physical  improba- 
bilities of  an  ellipsoidal  figure  are  so 
great  that  it  seems  more  reasonable  to  at- 
tribute any  apparent  departure  from  the 
spheroidal  figure  to  effects  of  local  at- 
traction. Again,  there  are  physical  rea- 
sons for  supposing  a  spheroidal  earth, 
but  the  existence  of  a  fluid  ellipsoid  can 
only  be  explained  by  supposing  the  exist- 
ence of  an  ellipsoidal  nucleus,  which  all 
speculators  in  cosmogony  agree  in  re- 
garding as  highly  improbable.  Arch. 
Pratt  remarks,  concerning  the  ellipsoidal 
figure,  that  "  if  a  very  large  number  of 
arcs  in  all  parts  of  the  world  were  meas- 
ured, and  local  attraction  being  taken  in- 


49 


to  account,  the  result  gave  an  ellipsoid 
with  its  two  equatorial  axes  differing  by  a 
quantity,  important  when  compared  with 
the  residual  errors  of  observation,  there 
might  be  some  argument  for  an  ellipsoid- 
al figure." 

During  the  present  century  the  ex- 
tensive trigonometrical  surveys  under- 
taken by  many  countries  have  been  the 
means  of  furnishing  a  number  of  long 
and  accurately  measured  arcs.  Of  these 
the  most  important  are  the  Anglo- Gallic, 
the  Russian,  the  Indian,  and  the  U.  IS. 
Coast  Survey.  The  first  three  have  been 
used  in  most  of  the  later  determinations 
of  the  mean  ellipse  mentioned  in  the 
table  on  page  44. 

Now,  it  is  highly  satisfactory  to  find 
that  the  oblate  spheroidal  figure,  thus 
practically  proved  to  exist,  is  what  theo- 
retically ought  to  result  from  the  rotation 
of  the  earth  on  its  axis.  The  form  of 
the  earth  is  owing  to  the  reciprocal  at- 
tractions of  its  component  particles. 
When  a  weight  is  whirled  around,  it  ac- 
quires a  tendency  to  recede  from  the  cen- 


50 


ter  of  its  motion,  as  a  stone  whirled 
around  in  a  sling.  This  tendency  is 
called  centrifugal  force.  Supposing  the 
rotation  of  the  earth,  a  centrifugal  force 
is  generated  whose  general  tendency  will 
be  to  cause  objects  to  fly  off  the  sur- 
face. This  force  diminishes  the  gravity 
of  particles,  and  hence  they  recede 
from  the  axis  of  the  earth  until,  by 
their  number  and  attraction,  they  counter- 
balance the  centrifugal  force.  This  is  con- 
firmed by  experience.  There  is  an  actual 
difference  in  the  force  of  gravity  or  down- 
ward tendency  of  the  same  body  when 
conveyed  successively  to  points  in  differ- 
ent latitudes.  Delicate  experiments,  con- 
ducted with  the  greatest  care,  have  .fully 
demonstrated  the  fact  of  a  regular  and 
progressive  increase  in  the  weight  of 
bodies,  corresponding  to  the  increase  of 
latitude. 

Now,  let  us  suppose  a  globe  of  the 
size  of  the  earth  to  be  uniformly  covered 
with  water.  So  long  as  the  body  re- 
mained fixed,  the  surface  of  the  water 
would  outline  a  perfect  sphere.  Imme- 


51 


diately  attending  the  introduction  of  ro- 
tation on  its  axis,  a  centrifugal  force 
would  be  developed  which  would  act  upon 
every  particle  in  such  a  way  as  to  tend  to 
cause  it  to  recede  from  the  axis  of  rota- 
tion. But  now  every  particle  would  be 
subject  to  the  action  of  two  forces — the 
one  just  mentioned,  and  that  of  gravity. 
The  direction  of  the  former  would  be 
perpendicular  to  the  axis  of  rotation,  and 
that  of  the  latter  perpendicular  to  the 
surface  of  the  water.  Since  at  no 
position  but  the  line  of  the  equator 
are  these  forces  directly  opposite, 
they  combine  to  form  a  third  force 
which  urges  every  particle  not  situated 
in  the  equator  towards  it  with  a  force  de- 
pendent upon  the  velocity  of  rotation. 
This  latter  force  and  the  figure  of  the  re- 
sulting surface  of  the  water  are  so  con- 
nected that  an  increase  of  centrifugal 
force  is  always  counterbalanced  by  a  pro- 
portionate change  in  the  direction  of  gra- 
vity. Therefore  the  water  would  recede 
from  the  poles  and  heap  itself  on  the 
equator.  This  would  leave  the  polar  re- 


52 


gions,  in  the  case  of  the  earth,  protuber- 
ant masses  of  land.  Now,  the  sea  is  con- 
stantly washing  and  grinding  away  the 
land,  and  carrying  and  depositing  pebbles 
and  fragments  over  its  bed.  Thus,  in 
the  case  considered,  the  water  beating 
the  polar  continents  would  gradually 
wear  them  down,  and,  as  with  the  mole- 
cules of  water,  so,  in  turn,  the  worn-off 
particles  and  fragments  of  the  polar  land 
would  would  be  forced  towards  the  equa- 
tor, till  the  earth  would  assume  by  de- 
grees the  form  we  have  shown  it  to  ap- 
proximate— the  oblate  spheroid. 


SECTION   II. 
THE    OBLATE 

SPHEROIDAL  HYPOTHESIS. 


SECTION  II. 


It  is  not  our  purpose  to  enter  into  the 
mathematical  discussion  of  the  fluid  the- 
ory of  the  Figure  of  the  Earth,  but  simply 
to  place  before  our  readers  such  princi- 
ples of  the  spheroidal  hypothesis  as  will 
lead  to  practical  results.  For  a  complete 
investigation  of  the  form  assumed  by  a 
revolving  fluid  on  the  principle  of  gravi- 
tation, we  would  refer  our  readers  to  sec- 
tion 2  of  Mr.  Airy's  essay  upon  the  Figure 
of  the  Earth  contained  in  the  "  Encyclo- 
pedia Metropolitans" 

We  have  stated  that  if  the  earth  be 
considered  a  fluid  mass,  the  form  of  the 
surface  will  be  an  oblate  spheroid  of 
small  ellipticity.  Further,  its  axis  will 
coincide  with  the  axis  of  revolution,  and 
the  surface  will  everywhere  be  perpendic- 
ular to  the  direction  of  gravity.  It  fol- 
lows also  upon  the  assumption  that  the 


56 


density  of  the  strata  varies  according  to 
a  certain  probable  law,  that  the  ellipticity 
is  jfa. 

Let  us  assume,  then,  that  the  mean 
Figure  of  the  Earth  is  an  oblate  spheroid, 
and  endeavor  to  show  by  what  methods 
an  ellipse  can  be  found  cutting  the  plumb 
line  at  right  angles  and  with  its  minor 
axis  coinciding  with  the  axis  of  the  earth. 
This  end  may  be  reached  by  four  methods, 
and  first  we  will  consider  how  the  Figure 
of  the  Earth  may  be  determined  from 
geodetic  operations.* 

*  We  do  not  purpose  to  deal  with  the  theory  of  the 
deduction  of  the  Figure  of  the  Earth  from  measure- 
ments of  the  Arcs  of  Longitude,  or  the  determinations 
of  Azimuths.  The  underlying  principles  of  these 
methods  have  had  a  very  limited  application ;  and, 
aside  from  the  Indian  Arc,  little  material  is  available. 
The  method  of  Azimuths  depends  entirely  upon  angles, 
and,  therefore,  can  only  assist  in  determining  the 
ellipticity. 


57 


CHAPTEK  I. 

THE  FIGURE  OF  THE  EARTH  DETERMINED  BY 
GEODETIC  OPERATIONS. 

The  first  step  in  this  method  is  to 
measure  as  accurately  as  possible  a  base 
line  of  any  convenient  length,  not  less 
than  5  or  6  miles,  and  as  near  as  possible 
to  the  meridian  upon  which  we  are  to 
base  our  calculations. 

For  this  purpose  wooden  or  metal 
rods,  steel  chains,  or  compensation  bars 
are  used.  The  latter  consist  of  a  series 
of  compound  bars,  self-correcting  for 
temperature.*  Two  bars,  one  of  brass, 
the  other  of  iron,  are  laid  side  by  side 
firmly  united  at  their  centers,  while  their 
ends  are  free  to  expand  or  contract. 
These  bars,  at  the  standard  temperature, 
are  of  the  same  length.  The  following 
is  the  principle  of  their  construction: 
Let  AB  be  one  bar,  A'B'  the  other. 
Draw  lines  through  similar  extremities 

*  Encyc.  Brit. 


58 


AA/  BB'  to  P  and  Q.  Make  A'P=B'Q, 
AA'=BB'.  Now  if  A'P  is  to  AP  as  the 
rate  of  expansion  of  the  bar  A'B'  is  to 
that  of  the  bar  AB,  then  the  distance  PQ 
will  be  nearly  invariable.  In  the  com- 
pleted instrument  P  and  Q  are  dots  10  feet 
apart.  In  the  measurement,  the  bars  when 
aligned  do  not  come  in  contact.  A  space 
of  six  inches  or  thereabouts,  is  left  be- 
tween each  bar  and  its  neighbor,  and  the 
interval  is  measured  by  an  ingenious 
micrometrical  device,  constructed  upon 
the  same  principle  as  the  bars  them- 
selves. 

The  United  States  Coast  Survey  in 
their  measurements,  generally  use  an 
apparatus  consisting  of  four  bars,  pro- 
tected by  a  wooden  covering.  The  bars 
are  placed  so  as  to  leave  between  them  a 
small  interval,  which  is  measured  by 
wires  brought  into  contact  and  adjusted 
by  means  of  micrometer  microscopes. 
The  bars  are  also  provided  with  ther- 
mometers. 

In  performing  the  operation  of  measur- 
ing the  base  it  is  very  important  that 


59 


at  least  two  of  the  rods  or  bars  should 
be  in  position  before  a  third  is  applied. 

The  base  being  thus  measured,  and 
the  reduction  for  inclination  applied 
together  with  the  correction  for  tem- 
perature, if  necessary,  there  re- 
mains but  to  reduce  the  length  to 
sea  level.  Let  r Dearth's  radius,  or  the 
radius  of  the  surface  or  the  sea,  and  h 
the  elevation ;  the  measured  lengths  must 

be  multiplied  by  the   fraction   -  — -   or 

1 — ,  in  order  to  obtain   the  length  at 

the  level  of  the  sea. 

Having  now  completed  the  operation 
known  as  the  measurement  of  the  base, 
we  next  proceed  to  measure  the  angles 
between  the  base  line,  and  visual  lines 
joining  the  extremities  of  the  base  to 
distant  points,  taken  as  near  .the  meridian 
as  convenient.  The  instruments  which 
have  been  used  for  measuring  these 
angles  are  quadrants,  theodolites  and  re- 
peating circles.  With  the  quadrant  and 
repeating  circle,  the  angle  actually  sub- 


60 


tended  by  the  points  or  signals  is  ob- 
served, while  with  theodolites  the  hori- 
zontal angle  is  measured.  Knowing  the 
length  of  one  side,  and  two  of  the 
angles  of  a  triangle,  we  can,  by  trigono- 
metrical formulae  deduce  the  lengths  of 
the  other  sides.  Repeating  the  operation 
with  the  sides  already  calculated,  and 
selecting  new  points  to  suit  the  emergen- 
cies of  the  case,  we  establish  a  connec- 
tion between  the  original  base  line  and  a 
second  base  at  the  termination  of  the 
chain  of  triangles,  and  obtain  the  length 
of  this  second  base  by  calculation.  It  is 
then  measured,  and  by  a  comparison  of 
the  calculated  and  measured  results  the 
correctness  of  the  operations  is  tested. 
The  next  step  is  to  determine  the  direc- 
tion of  one  of  fche  sides  of  the  chain  of 
triangles  with  regard  to  the  meridian. 
This  is  called  the  determination  of  the 
azimuth,  and  in  principle  is  as  follows— a 
temporary  mark  is  fixed  as  near  as 
possible  to  the  meridian ;  a  transit  in- 
strument is  adjusted  upon  it,  and  the 
transits  of  stars  at  different  polar  dis- 


61 

tances  are  observed.  The  deviation  of 
the  transit  instrument,  or  the  azimuth  of 
the  mark,  is  thus  found  with  great  accu- 
racy. By  means  of  a  theodolite  or  re- 
peating circle,  the  angle  between  the 
mark  and  one  of  the  signals  can  be  ob- 
served, and  there  results  the  azimuth  of 
the  station. 

This  having  been  satisfactorily  per- 
formed, we  are  now  able  to  calculate  the 
length  of  meridian  arc  contained  between 
the  two  parallels  passing  through  the  ex 
treme  stations  of  the  chain  of  triangles. 
The  simplest  means  "for  performing  this 
operation  is  known  as  the  method  of 
parallels  and  perpendiculars,  and  consists 
mainly  in  finding  the  projections  upon  the 
meridian  of  the  various  sides  of  the  system 
of  triangles.  Only  one  precaution  is  ne- 
cessary, and  that  is.  that  the  perpen- 
diculars to  the  meridian  must  not  be  of 
such  length  as  to  make  the  difference  be- 
tween the  spherical  length  of  the  sides 
and  their  length  when  projected  on  a 
tangent  plane — a  sensible  quantity. 


62 


63 


Let  the  accompanying  figure  represent 
a  chain  of  triangles,  in  which  A  and  G 
are  the  extreme  stations.  Let  AC  be  the 
base  line,  its  length  together  with  the 
azimuth  CAc  are  known,  and  therefore 
the  remaining  sides  and  their  projections 
upon  the  meridian  AB  maybe  calculated. 
The  sum  A#  of  the  projections  might  be 
supposed  to  represent  the  distance  on 
the  meridian  separating  A  and  G;  but 
this  is  not  the  case,  since  there  is  a 
difference  between  the  perpendicular  and 
the  arc  of  a  small  circle  passing  through 
G.  The  true  distance  is  A,/,  found  by 
producing  A#  to  B,  the  pole  of  the 
earth,  and  with  a  radius  BG  describing 
a  small  circle  Gg'.  The  difference  gg' 
of  the  two  distances  must  be  subtracted 
from  the  ascertained  length  A<7,  in  order 
to  obtain  the  true  distance  Kg'.  The 
quantity  gg'  may  be  found  as  follows : 

,_  (Gg.)*  tan,  lat.  G. 
.      gg  ~        earth's  diain. 

Another  method  considers  BG^  a  right- 
angled  spherical  triangle,  and  using  an 


64 


approximate  value  for  the  arc  of  a  great 
circle,  corresponding  to  the  length  Gg, 
determines  the  side  B<?  by 

cos.  BG 
Cos.  B#  =  - 

cos.  Gg 

If  we  join  B  and  G  by  the  arc  of  a 
great  circle,  it  is  evident  that  BG  is  not 
perpendicular  to  Gg,  and  therefore  not 
parallel  to  Bg.  This  is  what  is  under- 
stood as  the  convergence  of  the  meridians. 
The  angle  may  be  calculated  from  the 
spherical  triangle  BG<7,  as  follows  : 


The  determination  of  the  distance  -  be- 
tween the  parallels  completes  the 
Geodetic  operations,  and  we  have  now 
but  to  determine  the  latitudes  of  the  ex- 
tremities, an  operation  upon  the  accuracy 
of  which  the  whole  result  depends.  We 
will  not  attempt  a  description  of  the 
various  methods  employed,  but  deem 
it  sufficient  to  mention  the  application  of 
Airy's  Zenith  Sector. 

The  observation  consists  in  measuring 


65 


with  the  telescope  micrometer  the  differ- 
ence of  zenith  distances  of  two  stars 
which  cross  the  meridian,  one  to  the 
north,  and  the  other  to  the  south  of  the 
observer,  at  zenith  distances  which  differ 
by  not  much  more  than  10'  or  15',  the 
difference  of  the  times  of  transit  being 
not  less  than  one  or  more  than  twenty 
minutes.  The  peculiar  advantage  of  this 
method  is  that  refraction  is  well  nigh 
eliminated. 

Having  determined  the  latitudes  of  the 
two   extremities,    the   ellipse — of   which 
the  arc  is  part — is  found  as  follows  :* 
Let  I  and  I'  be  the  latitudes  of  the  ex- 
tremities of  the  arc. 

"     m  be  the  mean  of  I  and  I',  or  the 
middle  latitude. 

"     A  be  the  amplitude,  or  I—  I'. 

"     a  and  b  be  the  semi- axes. 

"     e  be  the  ellipticity. 

"     s  be  the  length  of  the  arc. 

"     r  be  the  radius  vector. 
Let  6  be  the  angle  r  makes  with  the  ma- 
jor axis. 

*  Pratt's  Figure  of  Earth. 


66 

Then 
1      cos.2#     sin.2<9 

•^=^-+-F-'    ^ 

L «2cos.2£+68  sin.8 


placing  b  =  a  (1  —  e) 
r=a(l  —  esin*l)  neglecting  £2, 

dr  .    ,        ,    dO 

in.2/ 


=  «(!— Je— f^cos.  20 

.-.  s=a[(l-±e)(l— I'}— ff(sin.2/— sin.2^')] 
=i  (a  +  6)  A— |  (a—b}  sin.  A,  cos.  2m.     (1) 

If  A  be  small,  not  exceeding  12°,  we 
may  place  sin.  A=A,  then 


when  the  lengths,   amplitudes  and  lati- 
tudes of  two  arcs  are  known. 


67 

s       s' 


a—b 


S  n  s'  r» 

,     —  cos.zm  —  —  cos.zm 
^±^_A_  A7  3) 

2  cos.  2  m'  —  cos.  2  m 

From  these  formulae  the  semi-axes, 
a  and  b,  may  be  computed  and  the  value 
for  e,  the  elliptic!  ty  follows. 

If  we  were  to  substitute  in  these  equa- 
tions observed  values  resulting  from  the 
various  measurements  of  meridian  arcs 
indifferent  parts  of  the  earth,  it  would 
be  found  that  the  calculated  semi-axes 
and  ellipticities  in  the  various  cases 
would  be  different.  If  the  figure  of  the 
earth  were  truly  spheroidal,  and  there" 
were  no  errors  in  the  data,  i.  e.,  in  the 
observed  amplitudes  and  measured  arcs, 
the  results  would  come  out  in  complete 
accordance  with  each  other.  But  this  is 
evidently  not  the  case. 

Let  us  inquire,  then,  as  to  the  source 
of  these  variations  in  the  calculated  spher- 
oid, and  in  attaining  this  end  we  will  first 


68 


state  the  assumptions  we  have  made  in 
deducing  the  above  formulae.  These  are 
(1)  that  the  meridian  arc  is  an.  ellipse  in 
accordance  with  the  fluid  theory.  (2) 
That  the  plumb  line  at  all  stations  is  nor- 
mal to  this  ellipse. 

It  is  plain  that  the  conclusions  of  the 
fluid  theory  are  not  applicable  to  the 
earth  in  its  present  state.  The  irregu- 
larities in  the  external  form  of  the  earth 
are  considerable  when  theoretically  con- 
sidered, and,  of  necessity,  these  irregu- 
larities of  the  surface  must  be  accom- 
panied by  irregularities  in  the  mathemati- 
cal figure  of  the  earth.  The  height  of 
mountains  and  depth  of  seas  are,  in  some 
places,  equal  to  \  the  difference  between 
the  equatorial  and  semi-polar  axes,  and 
we  are,  therefore,  prepared  to  admit  that 
the  surface  of  the  earth  is  not  one  of  rev- 
olution. Nevertheless,  there  must  be 
some  spheroid  that  agrees  very  closely 
with  the  mathematical  figure  of  the  earth 
and  having  the  same  axis  of  rotation. 

The  errors  due  to  local  attraction  re- 
sult from  the  deflection  of  the  plumb 


69 

line  from  the  normal  to  the  assumed  el- 
lipse. This  is  sufficient  in  certain  parts 
to  produce  material  errors  in  the  vertical, 
and  therefore  in  the  difference  of  latitude 
determined  by  the  zenith  distances  of 
stars. 

Thus,  suppose  A  and  B  to  be  two  sta- 
tions, and  that  at  A  there  is  a  disturbing 
force  drawing  the  plumb  line  through  an 
angle  tf ;  then  it  is  evident  that  the  ap- 
parent zenith  of  A  will  in  reality  be  that 
of  some  other  place  A',  whose  distance 
from  A  is  r#,  when  r  =  earth's  radius. 
Similarly,  if  there  is  a  disturbance  at  B 
of  the  amount  #',  the  apparent  zenith  of 
B  will  be  that  of  some  other  place  B', 
whose  distance  from  B  is  rd' '. 

Mountain  masses,  oceans,  and  varia- 
tions in  the  density  of  the  interior  of  the 
earth  are  the  chief  causes  of  local  attrac- 
tion. 

It  is  then  evident  that  our  assumptions 
in  the  preceding  formulae  are  at  fault,  and 
it  is  now  necessary  to  discover  some 
means  whereby  the  resulting  errors  may 
be  eliminated. 


70 


Bessel  was  the  first  to  devise  a  method 
whereby  the  results  of  all  the  surveys  in 
the  different  parts  of  the  earth  might  be 
brought  to  bear  simultaneously  upon 
this  problem.  He  was  followed  by  Capt. 
A.  R.  Clarke,  who  discussed  the  problem 
at  the  end  of  the  Ordnance  Survey  Vol- 
ume. 

Bessel' s  method  is  in  essence  as  fol- 
lows :  Corrections  expressed  in  algebraic 
terms  are  applied  to  the  latitudes  of  the 
several  stations,  dividing  the  arcs  into 
their  subordinate  parts,  such  as  to  make 
their  measured  lengths  exactly  fit  an  el- 
lipse. The  values  of  the  axes  of  this  el- 
lipse are  then  determined  so  as  to  make 
the  sum  of  the  squares  of  these  correc- 
tions a  minimum,  i.  e.,  an  ellipse  is  de- 
duced which  most  nearly  represents  the 
observations. 

We  will  now  endeavor  to  state  briefly 
how  this  may  be  done. 

Let  us  first  obtain  a  formula  for  cor- 
recting the  amplitude  of  an  arc  so  as  to 
make  its  measured  length  accord  with  a 
given  ellipse. 


71 


We  have  equation  (1), 

s=%  (a  +  b)  A— f  (a— b)  sin.  A  cos.  2  m. 

Suppose  now  that  x,  x'  are  small  cor- 
rections which  must  be  applied  to  the  ob- 
served latitudes  to  make  the  measured 
arc  fit  the  ellipse  of  which  a  and  b  are  the 
semi-axes ;  A  and  m  when  substituted  in 
the  above  formula  will  not  give  the  meas_ 
ured  value  of  s.  Instead  of  them 

X  +  x'—x  and  2m  +  x'  +  x 
must  be   substituted.     Hence   omitting 
very  small  quantities 

2s=  (a  +  b)  A  -  3  (a — b)  sin.  A  cos.  2m 
+  (xf— x)[a  +  b— 3(a— b)  cos.Acos.  2m\ 

/.  x' — x=( r— A  +  3 7  8in.Acos.2m\ 

\a  +  b  a  +  b  J 


(l  +  3^r 


cos.  A  cos.  2m       .     (4) 


Now  the  mean  radius  of  the  earth  is 
known  not  to  differ  much  from  20890000 
feet,  and  the  ellipticity  not  much  from 
•g-J-g-.  It  is  therefore  convenient  to  put 
a  +  b  under  the  form 


72 


a—b 


where  the  squares  of  u  and  v  may  be  neg- 
lected. 

Substituting  these  in  formula  (4)  and 
changing  the  form,  we  have 


x'= 

where  m,  a,  /?,  are  functions  of  the  ob- 
served latitudes,  the  measured  lengths 
and  other  numerical  quantities.* 

In  every  measured  arc,  not  only  are 
the  extreme  stations  determined  in  lati- 
tude, but  also  a  number  of  intermediate 
stations  ;  one  of  them  should  be  taken 
as  a  station  of  reference,,  and  each  separ- 
ate error  should  be  written  in  terms  of 
the  reference  station  error. 

*  The  values  of  m,  a  and  (3,  have  been  calculated  for 
the  principal  arcs  and  may  be  found  in  the  British 
Ordnance  Survey  volume. 


73 
Thus  if  we  have  five  stations, 


Now,  since  these  equations  are  affected 
by  errors  of  observation,  it  is  impossible 
to  find  values  of  u  and  v  which  will 
satisfy  all  the  equations,  and  therefore  it 
is  necessary  to  find  their  most  probable 
values.  According  to  the  method  of 
least  squares,  those  values  of  u  and  v 
are  the  most  probable,  which  render  the 
sum  of  the  squares  of  all  the  errors  a 
minimum. 

In  proceeding  with  this  method,  we 
first  deduce  normal  equations  for  u  by 
multiplying  each  observation  equation  by 
the  coefficient  of  u  in  that  equation,  and 
add  the  results ;  likewise,  deduce  normal 
equations  for  v  by  multiplying  each  ob- 
servation equation  by  the  coefficient  of  v 
in  that  equation,  and  add  the  results. 
Thus  we  will  have  two  normal  equations, 
each  containing  two  unknown  quantities, 


74 


and  the  solution  of  these  will  give  us  the 
most  probable  values  for  u  and  v.  Sub- 
stituting these  values  in  equations  (5),  (6), 
(7),  we  obtain  values  for  the  two  semi- 
axes  and  the  ellipticity. 

In  conclusion,  let  us  examine  what  ef- 
fect any  error  in  the  amplitudes  will 
have  upon  the  resulting  axes. 

If  we  differentiate  equations  (2,  3), 
there  will  appear  in  the  denominators  of 
the  resulting  expressions  the  quantity 

cos  2m — cos  2m'. 

The  errors  in  the  axes  dependent  upon 
the  amplitude  errors  will  therefore  be 
least  when  this  quantity  is  a  maximum 
If  an  arc  is  located  in  the  southern  half 
of  the  quadrant,  cos  2  m  is  positive, 
then, 

2m' =180°,  or  m'=90° 

will  give  the  best  result.  If  another 
arc  is  in  the  northern  half,  cos  2m  is  nega- 
tive, and 

cos  2m' =0 

will  give  the  best  result. 

Therefore  the  nearer  one  arc  is  to  the 


75 


pole,  and  the  other  to  the  equator,  the 
less  will  be  the  effect  upon  the  calcula- 
tions of  any  errors  in  the  observed  am- 
plitudes. 


76 


CHAPTER  II. 

THE  FIGURE  or  THE  EARTH  DEDUCED  FROM 
PENDULUM  EXPERIMENTS. 

Upon  the  hypothesis  of  the  earth  be- 
ing a  fluid  mass,  Clairaut  deduced  an 
equation  showing  that  the  increase  of 
gravity  in  passing  from  the  equator  to 
the  poles  varies  as  the  square  of  the  sine 
of  the  latitude,  and  that  a  certain  rela- 
tion must  necessarily  exist  between  the 
ellipticity  and  the  force  of  gravity. 
Therefore,  if  we  desire  to  use  this  the- 
orem in  the  determination  of  the  Figure 
of  the  Earth,  we  must  first  obtain  some 
practical  method  of  measuring  the  force 
of  gravity  in  any  latitude.  The  neces- 
sity of  using  the  pendulum  for  this  pur- 
pose will  easily  be  seen  if  we  consider 
the  impossibility  of  ascertaining  the 
magnitude  of  the  force  by  an  experiment 
upon  the  single  descent  of  free  bodies. 
The  quantity  to  be  measured  is  the  veloc- 
ity which  gravity  creates  in  any  freely 


77 


descending  body  by  its  action  continued 
during  a  second  of  time.  In  con- 
ducting a  series  of  experiments  it  is 
usual  to  observe  the  number  of  vibra- 
tions made  by  the  same  seconds'  pendu- 
lum in  the  different  places  at  which  it  is 
proposed  to  compare  the  force  of  grav- 
ity, and  likewise  the  number  of  vibrations 
made  at  London  or  Paris.  The  com- 
parative number  of  vibrations  being 
found,  the  comparative  force  of  grav- 
ity or  the  comparative  length  of  the  sec- 
onds' pendulum  can  be  deduced.  Since 
the  length  of  the  seconds'  pendulum  has 
been  very  accurately  determined  at  Lon- 
don and  Paris,  its  length  in  any  latitude 
may  be  found. 

In  order  to  understand  the  principles 
of  the  deduction  of  the  Figure  of  the 
Earth  from  pendulum  experiments,  let  us 
examine  the  theory  of  the  pendulum. 

In  the  following  investigation  the 
pendulum  is  taken  to  be  a  material  point 
suspended  by  a  string  without  weight — 
an  evident  impossibility.  It  may  be 
shown,  however,  that  if  a  compound 


78 


pendulum  be  given,  it  is  possible  to  calcu- 
late the  length  of  a  simple  pendulum 
with  the  same  vibration  period;  there- 
fore, what  is  true  of  the  imaginary 
pendulum,  is  true  of  the  pendulum  used 
in  practice,  and  the  latter  may  be  used 
for  all  the  purposes  of  the  former. 

It  is  known  that  the  time  of  vibration, 
in  seconds,  of  a  simple  pendulum,  whose 
length  is  I,  in  double  the  circular  arc, 
whose  versed  sign  is  b,  will  be  nearly 
expressed  by  ' 

* 


where  g  equals  the  space  through  which 
a  body  would  fall  in  a  second  of  time. 
If  the  arc  is  indefinitely  small,  the  time 
will  not  vary  much  from 


Now  let  us  assume  the  length  of 
pendulum,  such  that  the  time  of  vibra- 
tion shall  be  one  second,  or 


79 


The  space  g  then  results  from  multiply- 

2 

ing   the  length  of   the   pendulum  by  -^ 

Since  gravity  is  not  the  same  at  all 
places  on  the  earth's  surface,  the  length 
of  the  pendulum  vibrating  seconds  will 
vary.  However,  when  the  lengths  at  the 
different  places  are  determined,  the  same 
proportion  holds  between  those  lengths 
as  between  the  forces  of  gravity  at  the 
various  stations. 

Since  it  is  very  difficult  to  make  a 
pendulum  which  shall  vibrate  in  exactly 
one  second,  it  is  expedient  in  some  cases 
to  observe  the  time  in  which  a  pendulum 
of  known  length  vibrates,  and  to  deduce 
from  this  the  length  of  the  seconds 
pendulum.  In  the  expression 


where  t=  the  time  of  vibration,  it  is  evi- 
dent that  the  length  of  the  pendulum  is 
proportional  to  the  square  of  its  time  of 
vibration.  '  Thus  it  is  seen  that  the 
length  of  the  seconds'  pendulum  bears  to 


80 


the  known  length  the  same  ratio  as  unity 
to  the  square  of  the  number  of  seconds 
in  the  observed  time  of  vibration. 

In  this  method  the  absolute  length  of 
the  pendulum  is  determined  at  every 
station  of  observation,  and  while  in  some  - 
respects  this  is  preferable,  there  are  seri- 
ous disadvantages.  For  this  reason  a 
method  has  been  used  of  finding  the  pro- 
portion between  the  force  of  gravity  at 
each  station,  and  at  some  place  of  refer- 
ence where  its  absolute  length  has  been 
very  accurately  determined.  This  is 
done  by  "  transporting  the  same  pendu- 
lum, after  having  observed  it  at  the 
reference  station,  to  all  the  different 
stations,  and  observing  the  time  of  vibra- 
tion at  all." 

Since 


and  I  is  the  same  at  all  stations,  g  is 
inversely  proportional  to  t*.  Therefore, 
the  force  of  gravity  at  any  new  station  is 
to  the  force  of  gravity  at  the  place  of 
reference,  inversely,  as  the  square  of 


81 


the  time  of  vibration  at  the  new  station, 
to  the  square  of  the  time  of  vibration  at  the 
place  of  reference.  Or  again  directly  as 
the  square  of  the  number  of  vibrations 
per  day  at  the  new  station,  to  the  number 
of  vibrations  per  day  at  the  place  of  refer- 
ence. The  length  of  the  seconds'  pendu- 
lum being  in  the  same  ratio,  that  at  the 
new  station  may  be  deduced.  In  the  pre- 
ceding investigation  certain  suppositions 
are  made,  which  are  not  tenable  when 
the  nature  of  the  experiments  partake  of 
great  accuracy.  These  are,  namely — 1st, 
that  the  arcs  of  vibration  are  extremely 
small ;  2d,  that  the  length  of  the  pendu- 
lum continues  invariable  ;  3d,  that  the 
condition  of  the  atmosphere  is  the  same 
under  all  circumstances.  We  will  con- 
sider each  of  these  in  the  above  order. 

When  the  arc  of  vibration  is  so  large 
that  the  term  depending  upon  it  cannot 
be  neglected,  the  time  of  vibration  is  ex- 
pressed by 


If  the  arc  of  vibration  includes  n  de- 
grees on  each  side  of  the  vertical. 

07     .    2  n°       n3sin.al° 
0=1  versm  n  =21  sin.    ~^-~^  -  o  - 

2  2 

nearly  ;  and  the  time  of  vibration 


If  the  experiments  continue  for  so 
short  a  time  that  n  remains  practically 
invariable,  the  observed  time  of  vibration 
should  be  divided  by 

,    sin.3l° 

1  +  n--[6- 

or  the  number  of  vibrations  per  day 
should  be  multiplied  by  the  same 
quantity,  in  order  to  represent  the  time 
or  number  of  vibrations  in  indefinitely 
small  arcs.  When  the  observations  con- 
tinue for  a  long  period,  it  is  necessary  to 
know  what  is  the  law  of  decrease  of  the 
arc.  Borda  and  Airy  found,  by  direct  ex- 
periment, that  the  decreasing  arcs  form 
very  nearly  a  geometrical  series.  Let 
m=  the  number  of  vibrations  observed; 
and  p=  the  proportion  of  each  arc  to  the 


83 


preceding  arc,  where  1— p  isjvery  small. 
Let  m',  the  number  of  degrees  in  the  last 

m-l 

arc,  =np.      Now  the  lengths  of  the  suc- 
cessive arcs  are  n,  np, 

m-l 
np* np. 

The  sum  of  all  the  times  of  vibration 
are: 


16 


The  sum  of  the  degree  series  is  : 

l_»2m        l_p2ro-2 

-~—r=      *    a     nearly; 
1—  j92          1—  p* 

therefore  the  sum  of  all  the  times 


-l 


approx., 

o 

.-.  log.  /j2  =  —  (log.  n'—  log.  n) 


m 


84 


Now  log.  p*=  modulus  (p2— 1),  nearly, 
since  p*  —  1  is  very  small. 
2 

.'.    1  — #a  = =—, .(log.W-log.7l'). 

modulus  X  m  v 
Consequently  the  sum  of  all  the  times 


"  9      \  • 

modulus  X  (ri*— n'*)      sin.2l°  ) 

log.  n—  log.  n'      '       32       ); 
and  the  mean  of  the  observed  times 


modulus  X(^8-^/8)     sin.2  1°  ) 
log.  n—  log.  n'  32      ) 

The  quantity  within  the  brackets  is 
that  by  which  the  observed  mean  time  of 
vibration  should  be  divided,  or  the 
number  of  vibrations  per  day  should  be 
multiplied,  in  order  to  reduce  the  obser- 
vations to  vibrations  in  indefinitely  small 
arcs. 

In  the  second  place,  we  have  supposed 
the  length  of  the  pendulum  to  continue 
invariable.  It  is  well  known  that  a  me- 


85 

tallic  wire  or  bar  undergoes  considerable 
changes  in  its  length  with  changes  of 
temperature,  and  it  is  therefore  necessary 
to  reduce  the  number  of  vibrations  to 
the  number  which  would  have  been 
made  if  the  pendulum  had  remained  of 
constant  length.  Suppose  that  the 
pendulum's  length  is  increased  above 
that  at  the  standard  temperature  in  the 
ratio  of  l:l  +  y  (y  being  very  small.) 
Then  the  time  of  vibration  is  increased 

in  the   ratio   of    1:  VTTy,  or   1:  1+| 

nearly,  and  the  number  of  vibrations  per 
day  is  diminished  in  that  ratio.  There- 
fore the  observed  number  of  vibrations 
should  be  increased  in  this  ratio  in  order 
to  find  the  number  which  would  have 
been  made  if  the  pendulum  had  been 
of  constant  length.  The  same  law  holds 
if  the  temperature  were  decreased  in- 
stead of  increased. 

Thirdly,  we  have  supposed  that  the 
condition  of  the  atmosphere  remains  un- 
changed. The  vibrations  are  observed 
in  air,  and  since  the  state  of  the  air 


86 

is  variable  it  is  desirable  to  calculate  the 
number  of  oscillations  which  would  have 
been  made  in  a  vacuum.  Now  the  effect 
of  the  air  is  to  diminish  the  weight  of 
the  pendulum  by  a  quantity  equal  to  the 
weight  of  air  displaced.  If  this  diminu- 
tion be  in  the  ratio  of  1:1—2,  then  the 
time  of  vibration  (as  will  appear  by 
changing  g  into  g  (1— 2)  in  the  preceding 
expression),  will  be  increased  in  the  ratio 

of  1 :  — -==:  and  .'•.  the  number  of  vibra- 
tions per  day  diminished  in  the  ratio  of 
1 :  V 1— 2,  or  1:1— jr  nearly.  Therefore 

to  get  the  number  of  vibrations  in  a 
vacuum,  we  must  increase  the  observed 

number  in  the  ratio  of  1:1  +  ^. 

In  calculating  the  above  correction,  it 
is  necessary  to  observe  the  height  of  the 
barometer,  since  the  height  of  the  latter 
is  nearly  proportional  to  the  weight  of 
the  atmosphere. 

This  correction,  in  the  early  part  of  this 
century,  was  applied  for  the  reduction  of 


87 


the  observations  to  the  sea  level.  Dr, 
Young,  in  the  Phil.  Trans,  for  1819,  re- 
marked that  since  this  correction  neg- 
lected the  attraction  of  the  elevated  mass 
it  was  too  great.  He  adds  that,  assum- 
ing the  density  of  the  elevated  mass  to  be 

o  r 

—  the  mean  density  of  the  earth,  the  cor- 
5.5 

rection  would  be  .66,  that  of  the  usual 
reduction. 

Having  shown  how  refinement  of  ex- 
periment is  obtained,  we  will  now  en- 
deavor to  explain  by  what  means  we 
deduce  the  Figure  of  the  Earth  from 
Pendulum  Experiments. 

We  have  for  the  attraction  at  any  point 
of  a  spheroid  the  following  expression  : 


m,  in  the  earth,  =  -^^ 


In  which  m=  ratio  of  centrifugal  force  to> 
gravity,  /  and  g—  the  co-ordinates  of  the 


88 


point,  in  the  directions  of  CD  and  CE  re- 
spectively, b=  polar  axis,  £  =ellipticity  and 

*?.  ?-^=  gravity  at  the  equator. 
3       o 

For  the  force  at  the  pole  we  must  make 
f=o,  g—o,  and  the  expression  therefore 


For  the  force  at  the  equator,  we  must 
2  +  ^=aa=:52,   nearly  and  there- 
fore equatorial  gravity. 


The  excess  of  the  former  above  this  is 


and  the  ratio  of  this  excess  to  equatorial 

gravity  is  s  m—  f.     Let   this    =  w,   then 
Jj 

n-f-  f=-m,  or  we  have  deduced  Clairaut's 
theorem. 


IFlg.   3. 

Let  EF  represent  the  earth's  surface,  and 
let  PQ  be  the  normal  at  P. 

Then  P  Q  N  is  the  latitude  of  P  and 

QN=PQ,  cos.  I. 

Now  as  we  shall  have  to  substitute  only 
in  the  small  terms  of  the  equation,  PQ=& 
nearly,  and  QN=CN  nearly, 


nearly. 

Substituting  this  in  equation  (8),  we 
have  for  the  force  of  gravity 


.        e4. 

+ 


90 


Gravity,  therefore,  may  be  generally  ex- 
pressed by  the  formula 


,   ...     (9) 
where  E  =  equatorial  gravity  and 

5m 

n  +  e=-. 

Now  let  p  and  p'  be  the  lengths  of  the 
seconds  pendulum  in  latitudes  I  and  V, 
P  that  at  the  equator,  then  from  equation 
(9)  we  have 

p=P(l  +  n8m*l)  .     .     (10) 

p'=P(l  +  nsm.'l') 
where 


,.n= 


"—      *         2    Tt 

— sin.Y— sin.V 
P 

,  5m 

and  e=—  —n 


In  applying  the  preceding  equations  to 
a  series  of  pendulum  observations  for  the 
determination  of  the  Figure  of  the  Earth, 
the  principle  of  Least  Squares  is  applied 
as  in  the  last  chapter  to  a  number  of  ob- 


91 


servation  equations  of  the  form  (10),  there- 
by obtaining  the  most  probable  values  for 
P  and  71. 


92 


CHAPTER  III. 

1.  THE  ELLIPTICITY  OF  THE  EARTH  DEDUCED 

FROM  OBSERVED  INEQUALITIES  OF  THE 

MOON'S  MOTION. 

In  the  expression  for  the  tangent  of  th 
moon's  latitude  there  is  this  term* 

/      m\   4.(60)s  .  s 

—I  £-~n  I  •     \    -sm.  parallax.  — 
\       2  /     n3  TT  // 

sin.  obliquity,  cos.  obliquity,  sin.  6 

£      Earth's   Mass     70 
.Now  —  =  -  —  -  -  =r-=  -  =rr,  nearly, 
fA     Earth  +  Moon     71 

27.25 


The  mean  horizontal  parallax  =57', 

obliquity  =  23°—  28'  nearly, 

.-.  term=-/f-^)  .4891"  sin.  8 

Now  this  inequality  has  been  found  to 
exist  and  its  magnitude  has  been  inferred 
from  observation.  It  has  the  effect  of 
fncreasing  the  apparent  inclination  of  the 

*  Airy's  Math.  Tracts. 


93 

moon's  orbit  in  one  position  of  her  nodes 
and  diminishing  it  as  much  in  the  oppo- 
site position. 

It  is  found  by  observation  that  the  co- 
efficient =  -8" 

^=4frl=001635 


and      =  .001730 
2 


2.  THE  EARTH'S  ELLIPTICITY  DEDUCED  FROM 
THE  PRECESSION  OF  THE  EQUINOXES. 

The  formula  expressing  the  annual  Pre- 
cession is  the  following  : 


in  which  1=  the  obliquity  of  the  eclip- 
tic =  23°  —  28'—  18",  i  =  inclination  of 
moon's  orbit  to  ecliptic  =5°—  8'  —  50", 
n  and  n'  are  the  mean  motions  of  the 
earth  around  its  axis  and  around  the  sun 
and  their  ratio  =365.26,  n"  the  mean 
motion  of  the  moon  around  the  earth 


=  27.32  days,  v=  ratio  of  the  masses  of 
earth  and  moon  =75. 

Substituting  these  quantities 

C— A 

Annual  Precession  =  16225''.6 — ^ — 

where  A  and  C  are  the  principal  moments 
of  inertia  of  the  mass,  the  latter  about 
the  axis  of  revolution. 

Now  ^^=1.98177  (*-iro)'* 

.-.  Annual  Precession  =32155"  (e— %m) 
But  the  Precession  by  observation = 50".  1 

.-.  e— J  m  =  50.1-=-32155  =  0.0015581 
/.  £=0.0015581  +  00017271=^ 

In  estimating  the  reliance  to  be  placed 
on  these  results  it  must  be  observed  that 
unless  the  observations  extend  over  a 
period  greater  than  20  years  they  are  in- 
sufficient. This  renders  the  resulting  de- 
terminations less  reliable  than  they  other- 
wise would  be,  as  in  all  probability  the 
observations  which  are  compared  have 
been  made  by  different  persons  and  in 
different  manners.  The  small  lunar  in- 

*  See  Pratt's  Fig.  of  the  Earth.    Page  151. 


95 


equalities,  besides,  are  involved  among  a 
mass  of  terms  greater  than  themselves. 
Airy  remarks  concerning  this  fact  that  an 
error  in  their  determination  has  less  in- 
fluence on  the  value  of  £  than  an  equal 
error  in  the  determination  of  nutation. 
These  facts  show  clearly  that  the  deduc- 
tion of  the  two  preceding  divisions  cannot 
be  compared  with  those  of  geodetic  meas- 
ures and  pendulum  observations.  How- 
ever, the  close  agreement  of  the  results 
with  those  deduced  from  geodetic  meas- 
ures and  pendulum  experiments  is  sig- 
nificant. It  shows  that  in  the  main  the 
spheroidal  hypothesis  is  correct. 


*»*  Any  book  in  this  Catalogue  sent  free  by  mail  on  receipt  of  price. 

VALUABLE 

SCIENTIFIC    BOOKS 

PUBLISHED  BY 

D.  VAN  NOSTRAND, 

23  MURRAY  STREET  AND  27  WARREN  STREET,  N.  Y. 


ADAMS  (J.  W.)  Sewers  and  Drains  for  Populous  Districts. 
Embracing  Rules  and  Formulas  lor  the  dimensions  and 
construction  ol  works  of  Sanitary  Engineers.  Second  edi- 
tion. 8vo,  cloth $2  50 

ALEXANDER  (J.  H.)  Universal  Dictionary  of  Weights  and 
Measures,  Ancient  and  Modern,  reduced  to  the  standards 
of  the  United  States  of  America.  New  edition,  enlarged. 
8vo,  cloth „...  3  50 

ATWOOD  (GEO.)    Practical  Blow-Pipe  Assaying,  ismo,  cloth, 

illustrated 2  oo 

AUCHINCLOSS  (W.  S.)  Link  and  Valve  Motions  Simplified. 
Illustrated  with  37  wood-cuts  and  21  lithographic  plates, 
together  with  a  Travel  Scale  and  numerous  useful  tables. 
8vo,  cloth 3  OO 

AXON  (W.  E.  A.)  The  Mechanic's  Friend  :  a  Collection  of  Re- 
ceipts and  Practical  Suggestions  Relating  to  Aquaria- 
Bronzing— Cements— Drawing— Dyes— Electricity— Gilding 
— Glass-working  —  Glues  —  Horology  —  Lacquers— Locomo- 
tives— Magnetism — Metal-working  -Modelling —  Photogra- 
phy— Pyrotechny—  Railways— Solders— Steam-Engine—Tel- 
egraphy— Taxidermy — Varnishes — Waterproofing,  and  Mis- 
cellaneous Tools,  Instruments,  Machines,  and  Processes 
connected  with  tne  Chemical  and  Mechanic  Arts.  With  nu- 
merous diagrams  and  wood-cuts.  Fancy  cloth  150 

BACON  (F.  W)  A  Treatise  on  the  Richards  Steam-Engine 
Indicator,  with  directions  for  its  use.  By  Charles  T.  Por- 
ter. Revised,  with  notes  and  large  additions  as  developed 
by  American  practice ;  with  an  appendix  containing  useful 
formulae  and  rules  for  engineers .  Illustrated.  Third  edi- 
tion, izmo.cloth I  oo 


D.    VAN   NOSTBAND  S   PUBLICATIONS. 


BARBA  (J.)  The  Use  of  Steel  for  Constructive  Purposes  : 
Method  of  Working,  Applying,  and  Testing  Plates  and 
Brass.  With  a  Preface  by  A.  L.  Holley,  C.E.  i2mo,  cloth.fi  50 


BARNES  (Lt.  Com.  J.  S.,  U.  S.  N.)  Submarine  Warfare,  offen- 
sive and  defensive,  including  a  discussion  of  the  offensive 
Torpedo  System,  its  effects  upon  Iron-Clad  Ship  Systems 
and  influence  upon  future  naval  wars.  With  twenty  litho- 
graphic plates  and  many  wood-cuts.  8vo,  cloth  ............  5  oo 

BEILSTEIN  (F.)     An  Introduction  to  Qualitative  Chemical 

Analysis,  translated  by  I.  J.  Osbun.    i2mo,  cloth  ..........      75 

BENET  (Gen.  S.  V.,  U.  S.  A.)  Electro-Ballistic  Machines,  and 
the  Schultz  Chronoscope.  Illustrated.  Second  edition,  410, 
cloth  .......................................................  3  oo 

BLAKE  (W.  P.)  Report  upon  the  Precious  Metals  :  Being  Sta- 
tistical Notices  of  the  principal  Gold  and  Silver  producing 
regions  of  the  World,  represented  at  the  Paris  Universal 
Exposition.  8vo,  cloth  .....................................  2  oo 

-  Ceramic  Art.    A  Report  on  Pottery,   Porcelain,  Tiles, 
Terra  Cotta,  and  Brick.    8vo,  cloth  ........................  2  oo 

BOW  (R.  H.)     A  Treatise  on  Bracing,  with  its  application  to     £ 
Bridges  and  other  Structures  of  Wood  or  Iron.    156  illustra- 

•  tions.    8vo,  cloth  .........................................  150 


BOWSER  (Prof.  E.  A.)  An  Elementary  Treatise  on  Analytic 
Geometry,  embracing  Plane  Geometry,  and  an  Introduc- 
tion to  Geometry  of  three  Dimensions.  I2mo,  cloth I  75 


An  Elementary  Treatise  on  the  Differential  and  Integral 

Calculus.    With  numerous  examples.    I2mo,  cloth 2  25 

BURGH  (N.  P.)  Modern  Marine  Engineering,  applied  to  Pad- 
dle and  Screw  Propulsion.  Consisting  of  36  colored  plates, 
259  practical  wood-cut  illustrations,  and  403  pages  of  de- 
scriptive matter,  the  whole  being  an  exposition  of  the  pre- 
sent practice  of  Tames  Watt  &  Co.,  J.  &  G.  Rennie,  R.  Na- 
pier &  Sons,  and  other  celebrated  firms.  Thick  410  vol., 

cloth 1000 

Half  morocco 15  oo 

BURT  (W.  A.)  Key  to  the  Solar  Compass,  and  Surveyor's  Com- 
panion ;  comprising  all  the  rules  necessary  for  use  in  the 
field:  also  description  of  the  Linear  Surveys  and  Public 
Land  System  of  the  United  States,  Notes  on  the  Barome- 
ter, suggestions  for  an  outfit  for  a  survey  of  four  months, 
etc.  Second  edition.  Pocket-book  form,  tuck 2  50 

BUTLER  (Capt.  J.  S.,  U.  S.  A.)  Projectiles  and  Rifled  Cannon. 
A  Critical  Discussion  of  the  Principal  Systems  of  Rifling 
and  Projectiles,  with  Practical  Suggestions  for  their  Im- 
provement, as  embraced  in  a  Report  to  the  Chief  of  Ord- 
nance,  U.  S.  A.  36  plates.  410,  cloth 6  OO 


D.  VAN  NOSTRAND'S  PUBLICATIONS.  3 

CAIN  (Prof.  WM .)    A  Practical  Treatise  on  Voussoir  and  Solid 

and  Braced  Arches.    i6mo,  cloth  extra $i  75 

CALDWELL  (Prof.  GEO.  C.)  and  BRENEMAN  (Prof.  A.  A.) 
Manual  of  Introductory  Chemical  Practice,  for  the  use  of 
Students  in  Colleges  and  Normal  and  High  Schools.  Third 
edition,  revised  and  corrected.  8vo,  cloth,  illustrated.  New 
and  enlarged  edition I  5° 

CAMPIN  (FRANCIS).    On  the  Construction  of  Iron  Roofs.  8vo, 

with  plates,  cloth 2  oo 

CHAUVENET  (Prof.  W.)  New  method  of  correcting  Lunar 
Distances,  and  improved  method  of  finding  the  error  and 
rate  of  a  chronometer,  by  equal  altitudes,  ovo,  cloth 2  OO 

CHURCH  (JOHN  A.)    Notes  of  a  Metallurgical  Journey  in 

Europe.     8vo,  cloth 2  CO 

CLARK  (D.  KINNEAR,  C.E.)  Fuel:  Its  Combustion  and 
Economy,  consisting  of  Abridgments  of  Treatise  on  the 
Combustion  of  Coal  and  the  Prevention  of  Smoke,  by  C. 
W.  Williams  ;  and  the  Economy  of  Fuel,  by  T.  S.  Pri- 
deaux.  With  extensive  additions  on  recent  practice  in  the 
Combustion  and  Economy  of  Fuel :  Coal,  Coke,  Wood, 
Peat,  Petroleum,  etc.  i2mo,  cloth I  50 

A  Manual  of  Rules,  Tables,  and  Data  for  Mechanical 

Engineers.  Based  on  the  most  recent  investigations.     Illus- 
trated with  numerous  diagrams.     1,012  pages.    8vo,  cloth...  7  50 
Halfmorocco 1000 

CLARK  (Lt.  LEWIS,  U.  S.  N.)  Theoretical  Navigation  and 
Nautical  Astronomy.  Illustrated  with  41  wood-cuts.  8vo, 
cloth I  50 

CLARKE  (T.  C.)  Description  of  the  Iron  Railway  Bridge  over 
the  Mississippi  River  at  Quincy,  Illinois.  Illustrated  with 
21  lithographed  plans.  410,  cloth 75° 

CLEVENGER  (S.  R.)  A  Treatise  on  the  Method  of  Govern- 
ment Surveying,  as  prescribed  by  the  U.  S.  Congress  and 
Commissioner  of  the  General  Land  Office,  with  C9mplete 
Mathematical,  Astronomical,  and  Practical  Instructions  for 
the  use  of  the  United  States  Surveyors  in  the  field.  i6mo, 
morocco 2  50 

COFFIN  (Prof  J.  H.  C.)  Navigation  and  Nautical  Astrono- 
my. Prepared  for  the  use  of  the  U.  S.  Naval  Academy. 
Sixth  edition.  52  wood-cut  illustrations.  12010,  cloth 350 

COLBURN  (ZERAH).    The  Gas-Works  of  London.    I2mo, 

boards 6° 

COLLINS  (JAS.  E.)    The  Private  Book  of  Useful  Alloys  and 

Memoranda  for  Goldsmiths,  Jewellers,  etc.    i8mo,  cloth...      50 


4  D.  VAN  NOSTRAND'S  PUBLICATIONS. 

CORNWALL  (Prof.  H.  B.)  Manual  of  Blow  Pipe  Analysis, 
Qualitative  and  Quantitative,  with  a  Complete  System  of 
Descriptive  Mineralogy.  8vo,  cloth,  with  many  illustra- 
tions. N.Y.,  1882...  . $250 

CRAIG  (B.  F.)  Weights  and  Measures.  An  account  of  the 
Decimal  System,  with  Tables  of  Conversion  for  Commer- 
cial and  Scientific  Uses.  Square  32010,  limp  cloth 50 

CRAIG  (Prof.  THOS.)    Elements  of  the  Mathematical  Theory 

of  Fluid  Motion.    i6mo,  cloth 125 

DAVIS  (C.  B.)  and  RAE  (F.  B.)  Hand-Book  of  Electrical  Dia- 
grams and  Connections.  Illustrated  with  32  full-page  illus- 
trations. Second  edition.  Oblong  8vo,  cloth  extra 2  oo 

DIEDRICH  (JOHN).  The  Theory  of  Strains  :  a  Compendium 
for  the  Calculation  and  Construction  of  Bridges,  Roofs,  and 
Cranes.  Illustrated  by  numerous  plates  and  diagrams. 
8vo,  cloth 500 

DIXON  (D.  B.)  The  Machinist's  and  Steam-Fngineer's  Prac- 
tical Calculator.  A  Compilation  of  useful  Rules,  and  Prob- 
lems Arithmetically  Solved,  together  with  General  Informa- 
tion applicable  to  Shop-Tools,  Mill-Gearing,  Pulleys  and 
Shafts,  Steam-Boilers  and  Engines.  Embracing  Valuable 
Tables,  and  Instruction  in  Screw-cutting,  Valve  and  Link 
Motion,  etc.  i6mo,  full  morocco,  pocket  form  .  ..(In  press) 

DODD  (GEO.)  Dictionary  of  Manufactures,  Mining,  Ma- 
chinery, and  the  Industrial  Arts .  I2mo,  cloth I  50 

DOUGLASS  (Prof  S.  H.)  and  PRESCOTT  (Prof.  A.  B.)  Qual- 
itative Chemical  Analysis.  A  Guide  in  the  Practical  Study 
of  Chemistry,  and  in  the  Work  of  Analysis.  Third  edition. 
8vo,  cloth 3  50 

DUBOIS  (A.  J.)    The  New  Method  of  Graphical  Statics.  With 

60  illustrations.    8vo,  cloth i  50 

EASSIE  (P.  B.)  Wood  and  its  Uses.  A  Hand-Book  for  the  use 
of  Contractors,  Builders.  Architects,  Engineers,  and  Tim- 
ber Merchants.  Upwards  of  250  illustrations.  8vo,  cloth.  150 

EDDY  (Prof.  H.  T.)  Researches  in  Graphical  Statics,  embrac- 
ing New  Constructions  in  Graphical  Statics,  a  New  General 
Method  in  Graphical  Statics,  and  the  Theory  of  Internal 
Stress  in  Graphical  Statics.  8vo,  cloth i  50 

ELIOT  (Prof.  C.  W.)  and  STOKER  (Prof.  F.  H.)  A  Compen- 
dious Manual  of  Qualitative  Chemical  Analysis.  Revised 
with  the  co-operation  of  the  authors.  By  Prof.  William  R. 
Nichols.  Illustrated.  i2mo,  cloth i  50 

ELLIOT  (Maj.  GEO.  H.,  U.  S.  E.)  European  Light-House 
Systems .  Being  a  Report  of  a  Tour  of  Inspection  made  in 
1873.  51  engravings  and  21  wood-cuts.  8vo,  cloth 500 


D.    VAN  NOSTRAND  S   PUBLICATIONS.  5 

ENGINEERING  FACTS  AND  FIGURES.  An  Annual 
Register  of  Progress  in  Mechanical  Engineering  and  Con- 
struction for  the  years  1863-64-65-66-67-68.  Fully  illus- 
trated. 6  vols.  i8mo,  cloth  (each  volume  sold  separately), 
per  vol $2  50 

FANNING  (J.  T.)    A  Practical  Treatise  of  Water-Supply  En- 

fineering.     Relating  to  the  Hydrology,  Hydrodynamics,  and 
radical  Construction  of  Water-Works  in  North  America. 
Third  edition.      With  numerous   tables  and  180  illustra- 
tions.   650  pages.    8vo,  cloth 500 

FISKE  (BRADLEY  A..  U.  S.  N.)    Electricity  in  Theory  and 

Practice.    8vo,  cloth ....  2  50 

FOSTER  (Gen.  J.  G.,  U.  S.  A.)  Submarine  Blasting  in  Boston 
Harbor,  Massachusetts.  Removal  ot  Tower  and  Corwin 
Rocks.  Illustrated  with  seven  plates.  410,  cloth 3  50 

FOYE  (Prof.  J.  C.)  Chemical  Problems.  With  brief  State- 
ments of  the  Principles  involved.  Second  edition,  revised 
and  enlarged.  i6mo,  boards 50 

FRANCIS  (JAS.  B.,  C  E.)  Lowell  Hydraulic  Experiments : 
Being  a  selection  from  Experiments  on  Hydraulic  Motors, 
on  the  Flow  of  Water  over  Weirs,  in  Open  Canals  of  Uni- 
form Rectangular  Section,  and  through  submerged  Orifices 
and  diverging  Tubes.  Made  at  Lowell,  Massachusetts. 
Fourth  edition,  revised  and  enlarged,  with  many  new  ex- 
periments, and  illustrated  with  twenty-three  copperplate 
engravings.  410,  cloth 15  oo 

FREE-HAND  DRAWING.  A  Guide  to  Ornamental 
and  Landscape  Drawing.  By  an  Art  Student. 
boards 50 

GILLMORE  (Gen.  Q.  A.)  Treatise  on  Limes.  Hydraulic  Ce- 
ments, and  Mortars.  Papers  on  Practical  Engineering,  U.  . 
S.  Engineer  Department,  No.  9,  containing  Reports 'of 
numerous  Experiments  conducted  in  New  York  City  during 
the  years  1858  to  1861,  inclusive.  With  numerous  illustra- 
tions. 8vo,  cloth 400 

Practical  Treatise  on  the  Construction  of  Roads,  Streets, 

and  Pavements.    With  70  illustrations.     i2mo,  cloth 2  OO 

Report  on  Strength  of  the  Building  Stones  in  the  United 

States,  etc.    8vo,  illustrated,  cloth      150 

Coignet  Beton  and  other  Artificial  Stone.    9  plates,  views, 

etc.    8vo,  cloth 2  50 

GOODEVE  (T.  M.)    A  Text-Book  on  the  Steam-Engine.     143 

illustrations.     I2mo,  cloth 2  oo 

GORDON  (J.  E.  H.)    Four  Lectures  on  Static  Induction.    12010, 

cloth 80 


6  D.  VAN  NOSTRAND'S  PUBLICATIONS. 

GRUNER  (M.  L.)  The  Manufacture  of  Steel.  Translated 
from  the  French,  by  Lenox  Smith,  with  an  appendix  on  the 
Bessemer  process  in  the  United  States,  by  the  translator. 
Illustrated.  8vo,  cloth $3  50 

HALF-HOURS  WITH  MODERN  SCIENTISTS.  Lectures 
and  Essays.  By  Professors  Huxley,  Barker,  Stirling,  Cope, 
Tyndall,  Wallace,  Roscoe,  Hoggins,  Lockyer,  Young, 
Mayer,  and  Reed.  Being  the  University  Series  bound  up. 
With  a  general  introduction  by  Noah  Porter,  President  of 
Yale  College.  2  vols.  i2mo,  cloth,  illustrated 2  50 

HAMILTON  (W.  G.)  Useful  Information  for  Railway  Men. 
Sixth  edition,  revised  and  enlarged.  562  pages,  pocket  form. 
M orocco,  gilt 7 2  oo 

HARRISON  (W.  B.)  The  Mechanic's  Tool  Book,  with  Prac- 
tical Rules  and  Suggestions  for  Use  of  Machinists,  Iron- 
Workers,  and  others.  Illustrated  with  44  engravings. 
I2mo,  cloth ™ j  50 

HASKINS  (C.  H.)  The  Galvanometer  and  its  Uses.  A  Man- 
ual  for  Electricians  and  Students.  Second  edition.  i2mo, 
morocco i  50 

HENRICI  (OLAUS).  Skeleton  Structures,  especially  in  their 
application  to  the  Building  of  Steel  and  Iron  Bridges.  With 
folding  plates  and  diagrams.  8vo,  cloth I  50 

HEWSON  (WM.)  Principles  and  Practice  of  Embanking 
Lands  from  River  Floods,  as  applied  to  the  Levees  of  the 
Mississippi.  8vo,  cloth 2  oo 

HOLLEY  (ALEX.  L.)  A  Treatiseon  Ordnance  and  Armor,  em- 
bracing descriptions,  discussions,  and  professional  opinions 
concerning  the  materials,  fabrication,  requirements,  capa- 
bilities, and  endurance  of  European  and  American  Guns, 
for  Naval,  Sea-Coast,  and  Iron-Clad  Warfare,  and  their 
Rifling,  Projectiles,  and  Breech-Loading;  also,  results  of 
experiments  against  armor,  from  official  records,  with  an 
appendix  referring  to  Gun-Cotton,  Hooped  Guns,  etc.,  etc. 
948  pages,  493  engravings,  and  147  Tables  of  Results,  etc. 
8vo,  half  roan 10  oo 

— —  Railway  Practice  American  and  European  Railway 
Practice  in  the  economical  Generation  of  Steam,  including 
the  Materials  and  Construction  of  Coal-burning  Boilers, 
Combustion,  the  Variable  Blast,  Vaporization,  Circulation, 
Superheating,  Supplying  and  Heating  Feed-water^  etc., 
and  the  Adaptation  of  Wood  and  Coke-burning  Engines  to 
Coal-burning ;  and  in  Permanent  Way,  including  Road-bed, 
Sleepers,  Rails,  Joint-fastenings,  Street  Railways,  etc.,  etc. 
With  77  lithographed  plates.  Folio,  cloth 12  CO 

HOWARD  (C.  R.)  Earthwork  Mensuration  on  the  Basis  of 
the  Prismoidal  Formulae.  Containing  simple  and  labor- 
saving  method  of  obtaining  Prismoidal  Contents  directly 


D.    VAX  NOSTRAND  S   PUBLICATIONS.  7 

from  End  Areas.  Illustrated  by  Examples,  and  accom- 
panied by  Plain  Rules  for  Practical  Uses.  Illustrated.  8vo, 
cloth  $i  50 

INDUCTION-COILS.  How  Made  and  How  Used.  63  illus- 
trations. i6mo,  boards  50 

ISHERWOOD  (B.  F.)  Engineering  Precedents  for  Steam  Ma- 
chinery. Arranged  in  t_ne  most  practical  and  useful  manner 
for  Engineers.  With  illustrations.  Two  volumes  in  one. 
8vo,  cloth 250 

JANNETTAZ  (EDWARD).  A  Guide  to  the  Determination  of 
Rocks:  being  an  Introduction  to  Lithology.  Translated 
from  the  French  by  G.  W.  Plympton,  Professor  of  Physical 
Science  at  Brooklyn  Polytechnic  Institute,  izmo,  cloth....  i  50 

JEFFERS  (Capt.  W.  N..  U.  S.  N.)  Nautical  Surveying.  Illus- 
trated witn  9  copperplates  and  31  wood-cut  illustrations. 
8vo,  cloth 5  oo 

JONES  .(H.  CHAPMAN).  Text-Book  of  Experimental  Or- 
ganic Chemistry  for  Students.  i8mo,  cloth I  co 

JOYNSON  (F.  H.)  The  Metals  used  in  Construction:  Iron, 

Steel,  Bessemer  Metal,  etc.,  etc.  Illustrated.  I2mo,  cloth.  75 

Designing  and  Construction  of  Machine  Gearing.  Illus- 
trated .  8vo,  cloth 2  oo 

KANSAS  CITY  BRIDGE  (THE).  With  an  account  of  the 
Regimen  of  the  Missouri  River,  and  a  description  of  the 
methods  used  for  Founding  in  that  River.  By  O.  Chanute, 
Chief-Engineer,  and  George  Morrison,  Assistant-Engineer. 
Illustrated  with  five  lithographic  views  and  twelve  plates  of 
plans.  410,  cloth 600 

KING  (W.  H.)  Lessons  and  Practical  Notes  on  Steam,  the 
Steam-Engine,  Propellers,  etc.,  etc.,  for  young  Marine  En- 
gineers, Students,  and  others.  Revised  by  Chief-Engineer 
T.  W.  King,  U.  S.  Navy.  Nineteenth  edition,  enlarged. 
8vo,  cloth 2  oo 

KIRKWOOD  (JAS.  P.)  Report  on  the  Filtration  of  River 
Waters  for  the  supply  of  Cities,  as  practised  in  Europe, 
made  to  the  Board  of  Water  Commissioners  of  the  City  of 
St.  Louis.  Illustrated  by  30  double-plate  engravings.  410, 
cloth 1500 

LARRABEE  (C.  S.)  Cipher  and  Secret  Letter  and  Telegra- 
phic Code,  with  Hogg's  Improvements.  The  most  perfect 
secret  code  ever  invented  or  discovered.  Impossible  to  read 
without  the  key.  i8mo,  cloth I  oo 

LOCK  (C.  G.),  WIGNER  (G.  W.),  and  HARLAND  (R.  H.) 
Sugar  Growing  and  Refining.  Treatise  on  the  Culture  of 
Sugar- Yielding  Plants,  and  the  Manufacture  and  Refining  of 
Cane,  Beet,  and  other  sugars.  8vo,  cloth,  illustrated 12  oo 


8  D.  VAN  NOSTRAND'S  PUBLICATIONS. 

LOCKWOOD  (THOS.  D.)  Electricity.  Magnetism,  and  Elec- 
tro-Telegraphy. A  Practical  Guide  for  Students,  Operators, 
and  Inspectors.  8vo,  cloth $2  50 

LORING  (A.  E.)  A  Hand-Book  on  the  Electro-Magnetic  Tele- 
graph. Paper  boards 50 

Clot  h 75 

Morocco I  oo 

MAcCORD  (Prof.  C.  W.)     A  Practical  Treatise  on  the  Slide- 
Valve  by  Eccentrics,  examining  by  methods  the  action  of 
the  Eccentric  upon  the  Slide-Valve,  and  explaining  the  prac- 
tical processes  of  laying  out  the  movements,  adapting  the  . 
valve  for  its  various  duties  in  the  steam-engine.     Second  edi*1.' 
tion.    Illustrated.     4to,  cloth '!....?»  50 

McCULLOCH  (Prof.  R   S.)    Elementary  Treatise  on  the;Me-  ' 
chanical  Theory  of  Heat,   and  its  application  to  Air* and 
Steam  Engines.     8vo,  cloth 3  50 

MERRILL  (Col.  WM.  E.,  U.  S.  A.)    Iron-Truss  Bridged  for  • 
Railroads.     The  method  of  calculating  strains  in  Trusses, 
with  a  careful  comparison  of  the  most  prfiminent  Trusses,  in    , 
reference  to  economy  in  combination,  etc.^»etc.     Illustrated.    ' 
4to,  cloth 5*00 

MICHAELIS  (Capt.   O.  E.,    U.    S.   A.)     The  Le   Boufenge 
Chronograph,  with  three  lithograph  folding  plates  of  illus-     •. 
trations.    4to,  cloth « 3  fO 

MICHIE  (Prof.  P.  S.)     Elements  of  Wave  Motion  relating^to 
Sound  and  Light.     Text-Book  forthe  U.S.  Military  Acade-      ' 
my.    8vo,  cloth,  illustrated \. .   5  *O 

MINIFIE(WM.)  Mechanical  Drawing.  A  Text-Book  of  Geo-  , 
metrical  Drawing  for  the  use  of  Mechanics  and  Schools,  in 
which  the  Definitions  and  Rules  of  Geometry  are  familiarly 
explained  ;  the  Practical  Problems  are  arranged,  from  the 
most  simple  to  the  more  complex,  and  in  their  description 
technicalities  are  avoided  as  much  as  possible.  With  illus- 
trations for  Drawing  Plans,  Sections,  and  Elevations  of 
Railways  and  Machinery;  an  Introduction  to  Isometrical 
Drawing,  and  an  Essay  on  Linear  Perspective  and  Shadows. 
Illustrated  with  over  200  diagrams  engraved  on  steel.  _  Ninth 
edition.  With  an  Appendix  on  the  Theory  and  Application 

of  Colors.    8vo,  cloth  4  oo 

"  It  is  the  best  work  on  Drawing  that  we  have  ever  seen,  and  is 
especially  a  text-book  of  Geometrical  Drawing  for  the  use  of  Me- 
chanics and  Schools.  No  young  Mechanic,  such  as  a  Machinist, 
Engineer,  Cabinet-maker,  Millwright,  or  Carpenter,  should  be  with- 
out it." — Scientific  American. 

Geometrical  Drawing.  Abridged  from  the  octavo  edi- 
tion, for  the  use  of  schools.  Illustrated  with  forty-eight, 
steel  plates.  Fifth  edition.  i2mo,  cloth  200 


to  Circulation  Desk. 
Renewals  and  recharges  may  be  made  4  days  prior 

to  due  date. 

ALL  BOOKS  ARE  SUBJECT  TO  RECALL  7  DAYS 
AFTER  DATE  CHECKED  OUT. 


JJL 


LD21 — A-40m-8,'75 
(S7737L) 


General  Library 

University  of  California 

Berkeley 


